|
|
| User: "Richard Henry" |
|
| Title: Re: equal temperance vs. pure temperance music frequency ratios |
12 Sep 2003 12:24:02 PM |
|
|
"Tim923" <4-club-juggler@verizon.net> wrote in message
news:qfr3mvkujoga0b22bs8m24nv1n0v8arvvc@4ax.com...
If we designed the piano to play the key of C, using pure intervals
with no temperment, and C gets the frequency 1:
1st - 1
2nd - ?
3rd - 1.25
4th - 1.333...
5th - 1.5
6th - 1.666...
7th - ?
octave - 2
black keys:
minor 2nd - ?
minor 3rd - 1.2
minor 5th - ?
minor 6th - 1.6
minor 7th - ?
12th root of 2 is
1.0594630943592952645618252949463
(caution - results obtained with windows calculator)
so the first step (C#) should be 1.0594630943592952645618252949463
C - 1
C# - 1.0594630943592952645618252949463
D - 1.1224620483093729814335330496792
D# - 1.1892071150027210667174999705605
E - 1.2599210498948731647672106072782
F - 1.3348398541700343648308318811845
F# - 1.4142135623730950488016887242097
G - 1.4983070768766814987992807320298
G# - 1.5874010519681994747517056392715
A - 1.6817928305074290860622509524664
A# - 1.7817974362806786094804524111801
B - 1.8877486253633869932838263133341
C - 1.9999999999999999999999999999989
.
|
|
|
| User: "Harry Conover" |
|
| Title: Re: equal temperance vs. pure temperance music frequency ratios |
12 Sep 2003 07:08:59 PM |
|
|
"Richard Henry" <rphenry@home.com> wrote in message news:<Uin8b.848$v%5.486@fed1read02>...
"Tim923" <4-club-juggler@verizon.net> wrote in message
news:qfr3mvkujoga0b22bs8m24nv1n0v8arvvc@4ax.com...
If we designed the piano to play the key of C, using pure intervals
with no temperment, and C gets the frequency 1:
1st - 1
2nd - ?
3rd - 1.25
4th - 1.333...
5th - 1.5
6th - 1.666...
7th - ?
octave - 2
black keys:
minor 2nd - ?
minor 3rd - 1.2
minor 5th - ?
minor 6th - 1.6
minor 7th - ?
12th root of 2 is
1.0594630943592952645618252949463
(caution - results obtained with windows calculator)
so the first step (C#) should be 1.0594630943592952645618252949463
C - 1
C# - 1.0594630943592952645618252949463
D - 1.1224620483093729814335330496792
D# - 1.1892071150027210667174999705605
E - 1.2599210498948731647672106072782
F - 1.3348398541700343648308318811845
F# - 1.4142135623730950488016887242097
G - 1.4983070768766814987992807320298
G# - 1.5874010519681994747517056392715
A - 1.6817928305074290860622509524664
A# - 1.7817974362806786094804524111801
B - 1.8877486253633869932838263133341
C - 1.9999999999999999999999999999989
Sorry Richard, no. The 12th root of 2 comes into play only in equally
tempered note systems, not in the naturally tempered system that the
poster in asking about.
The difference is subtle, but very important. Equal note spacing allow
an instrument to played any key, but a natural temperment instrument
can only play in certain keys lest the "Wolf" in it be revealed
(particularly organs)
http://www.fictionwise.com/ebooks/eBook16257.htm
I wish I had a better reference, but Google let me down a bit on "Wolf
in an organ" search. Still, the citation I posted gives you the
general idea.
Harry C.
.
|
|
|
| User: "Richard Henry" |
|
| Title: Re: equal temperance vs. pure temperance music frequency ratios |
12 Sep 2003 07:37:08 PM |
|
|
"Harry Conover" <hhc314@yahoo.com> wrote in message
news:7ce4e226.0309121608.3c9eeb68@posting.google.com...
"Richard Henry" <rphenry@home.com> wrote in message
news:<Uin8b.848$v%5.486@fed1read02>...
"Tim923" <4-club-juggler@verizon.net> wrote in message
news:qfr3mvkujoga0b22bs8m24nv1n0v8arvvc@4ax.com...
If we designed the piano to play the key of C, using pure intervals
with no temperment, and C gets the frequency 1:
1st - 1
2nd - ?
3rd - 1.25
4th - 1.333...
5th - 1.5
6th - 1.666...
7th - ?
octave - 2
black keys:
minor 2nd - ?
minor 3rd - 1.2
minor 5th - ?
minor 6th - 1.6
minor 7th - ?
12th root of 2 is
1.0594630943592952645618252949463
(caution - results obtained with windows calculator)
so the first step (C#) should be 1.0594630943592952645618252949463
C - 1
C# - 1.0594630943592952645618252949463
D - 1.1224620483093729814335330496792
D# - 1.1892071150027210667174999705605
E - 1.2599210498948731647672106072782
F - 1.3348398541700343648308318811845
F# - 1.4142135623730950488016887242097
G - 1.4983070768766814987992807320298
G# - 1.5874010519681994747517056392715
A - 1.6817928305074290860622509524664
A# - 1.7817974362806786094804524111801
B - 1.8877486253633869932838263133341
C - 1.9999999999999999999999999999989
Sorry Richard, no. The 12th root of 2 comes into play only in equally
tempered note systems, not in the naturally tempered system that the
poster in asking about.
The difference is subtle, but very important. Equal note spacing allow
an instrument to played any key, but a natural temperment instrument
can only play in certain keys lest the "Wolf" in it be revealed
(particularly organs)
http://www.fictionwise.com/ebooks/eBook16257.htm
I wish I had a better reference, but Google let me down a bit on "Wolf
in an organ" search. Still, the citation I posted gives you the
general idea.
Actually, I ws surprised that a chain of windows calculator calculations
could come out so close to 2. By the way, there is a subtle error in the
list of numbers I posted above, for the real calculator freaks.
Almost on topic: do you have a good reference to an online virtual
synthesizer keyboard, where clicking or moving the cursor over a key will
result in the selected tone being heard?
.
|
|
|
| User: "Lucius Chiaraviglio" |
|
| Title: Re: equal temperance vs. pure temperance music frequency ratios |
01 Oct 2003 03:27:47 PM |
|
|
On Fri, 12 Sep 2003 17:37:08 -0700, Richard Henry wrote:
[. . .]
Almost on topic: do you have a good reference to an online virtual
synthesizer keyboard, where clicking or moving the cursor over a key will
result in the selected tone being heard?
This isn't in keyboard format, and it requires Java, but check out
http://pages.globetrotter.net/roule/accord.htm (JavaTuner).
--
Lucius Chiaraviglio
Approximate E-mail address:
To get the exact address: ^^^ ^replace this with 'r'
|||
replace this with single digit meaning the same thing
(Spambots of Doom, take that!).
.
|
|
|
|
|
|
|
| User: "Eric Prebys" |
|
| Title: Re: equal temperance vs. pure temperance music frequency ratios |
12 Sep 2003 12:24:52 PM |
|
|
Tim923 wrote:
If we designed the piano to play the key of C, using pure intervals
with no temperment, and C gets the frequency 1:
1st - 1
2nd - ?
3rd - 1.25
4th - 1.333...
5th - 1.5
6th - 1.666...
7th - ?
octave - 2
black keys:
minor 2nd - ?
minor 3rd - 1.2
minor 5th - ?
minor 6th - 1.6
minor 7th - ?
If you start at a particular note, you can generate a
untempered major scale by making a "circle of fifths" (i.e.
going up 3/2 in frequency) and then dropping back (dividing
by 2) when necessary to stay in a single octave.
Interestingly, if you
start on A, what you'll get is an E-major scale. I
think that long ago they made the choice to change it
because the fourth is weird otherwise, but maybe a music
expert can comment on that. Anyway, if you
keep going on this circle of fifths, you'll eventually
fill in the chromatic notes, and arrive *almost* back
to the octave after a total of 12.
I have a brief description of this (with all the notes)
in Appendix C of the following:
http://viper.princeton.edu/~prebys/p104_s01/satlab/satlab2.html
which I wrote for a lab I was teaching once.
-Eric
.
|
|
|
| User: "Lucius Chiaraviglio" |
|
| Title: Re: equal temperance vs. pure temperance music frequency ratios |
01 Oct 2003 04:26:37 PM |
|
|
Sorry about the botched multiple posts -- it looks like I got hit by
"nasty bug that could post the same article more than once" (in pan
0.14.0.94, as noted in http://pan.rebelbase.com). It looks like I also
got hit by a bug in the automatic text reformatting routine. And now it
looks like my news server is doing weird stuff. So here's the 3rd attempt
to fix the botched text:
To see what I mean, paste the following lines into Gnumeric or Excel:
1 =ln(A1)/ln(2) =round(B1) =2^C1 =A1-D1 =E1/D1
=A1*1.5 =ln(A2)/ln(2) =round(B2) =2^C2 =A2-D2 =E2/D2
and then replicate the second line (allowing your spreadsheet to increment
the line numbers normally) down to at least line 42. At G3 tack on:
=2^(1/6) =(G3-F3)-1
and at H42 tack on:
=F42/42
and then compare H3 to H42.
--
Lucius Chiaraviglio
Approximate E-mail address:
To get the exact address: ^^^ ^replace this with 'r'
|||
replace this with single digit meaning the same thing
(Spambots of Doom, take that!).
.
|
|
|
|
| User: "Dirk Van de moortel" |
|
| Title: Re: equal temperance vs. pure temperance music frequency ratios |
12 Sep 2003 01:11:57 PM |
|
|
"Eric Prebys" <prebys@fnal.gov> wrote in message news:3F620164.6070002@fnal.gov...
Tim923 wrote:
If we designed the piano to play the key of C, using pure intervals
with no temperment, and C gets the frequency 1:
1st - 1
2nd - ?
3rd - 1.25
4th - 1.333...
5th - 1.5
6th - 1.666...
7th - ?
octave - 2
black keys:
minor 2nd - ?
minor 3rd - 1.2
minor 5th - ?
minor 6th - 1.6
minor 7th - ?
If you start at a particular note, you can generate a
untempered major scale by making a "circle of fifths" (i.e.
going up 3/2 in frequency) and then dropping back (dividing
by 2) when necessary to stay in a single octave.
Interestingly, if you
start on A, what you'll get is an E-major scale. I
think that long ago they made the choice to change it
because the fourth is weird otherwise, but maybe a music
expert can comment on that. Anyway, if you
keep going on this circle of fifths, you'll eventually
fill in the chromatic notes, and arrive *almost* back
to the octave after a total of 12.
I have a brief description of this (with all the notes)
in Appendix C of the following:
http://viper.princeton.edu/~prebys/p104_s01/satlab/satlab2.html
which I wrote for a lab I was teaching once.
-Eric
Very nice and interesting lab and exactly the answer
the OP needs :-)
Btw, Eric, there's a little typo near the end:
| Based on this scale, our A-major chord becomes
| 440, 554.37, and 559.26 Hz ...
The last frequency should be 659.26
Cheers,
Dirk Vdm
.
|
|
|
| User: "Eric Prebys" |
|
| Title: Re: equal temperance vs. pure temperance music frequency ratios |
12 Sep 2003 01:58:17 PM |
|
|
Dirk Van de moortel wrote:
"Eric Prebys" <prebys@fnal.gov> wrote in message news:3F620164.6070002@fnal.gov...
Tim923 wrote:
If we designed the piano to play the key of C, using pure intervals
with no temperment, and C gets the frequency 1:
1st - 1
2nd - ?
3rd - 1.25
4th - 1.333...
5th - 1.5
6th - 1.666...
7th - ?
octave - 2
black keys:
minor 2nd - ?
minor 3rd - 1.2
minor 5th - ?
minor 6th - 1.6
minor 7th - ?
If you start at a particular note, you can generate a
untempered major scale by making a "circle of fifths" (i.e.
going up 3/2 in frequency) and then dropping back (dividing
by 2) when necessary to stay in a single octave.
Interestingly, if you
start on A, what you'll get is an E-major scale. I
think that long ago they made the choice to change it
because the fourth is weird otherwise, but maybe a music
expert can comment on that. Anyway, if you
keep going on this circle of fifths, you'll eventually
fill in the chromatic notes, and arrive *almost* back
to the octave after a total of 12.
I have a brief description of this (with all the notes)
in Appendix C of the following:
http://viper.princeton.edu/~prebys/p104_s01/satlab/satlab2.html
which I wrote for a lab I was teaching once.
-Eric
Very nice and interesting lab and exactly the answer
the OP needs :-)
Btw, Eric, there's a little typo near the end:
| Based on this scale, our A-major chord becomes
| 440, 554.37, and 559.26 Hz ...
The last frequency should be 659.26
Thanks for pointing that out. I'll try to fix it
at some point.
-Eric
Cheers,
Dirk Vdm
.
|
|
|
|
|
| User: "Martin Hogbin" |
|
| Title: Re: equal temperance vs. pure temperance music frequency ratios |
12 Sep 2003 03:29:08 PM |
|
|
"Eric Prebys" <prebys@fnal.gov> wrote in message news:3F620164.6070002@fnal.gov...
Tim923 wrote:
If we designed the piano to play the key of C, using pure intervals
with no temperment, and C gets the frequency 1:
1st - 1
2nd - ?
3rd - 1.25
Better to use fractions rather than decimals. The major scale notes
are then given by ratios of small integers.
2nd 9:8
3rd 5:4
4th 4:3
5th 3:2
6th 5:3
7th 15:8
8ve 2:1
This scale has, so called, just intonation.
If you start at a particular note, you can generate a
untempered major scale by making a "circle of fifths" (I.a.
going up 3/2 in frequency) and then dropping back (dividing
by 2) when necessary to stay in a single octave.
The scale you get by this method is not just intonation.
As Eric says below, it does not quite get back to its
starting point.
Interestingly, if you
start on A, what you'll get is an A-major scale. I
think that long ago they made the choice to change it
because the fourth is weird otherwise, but maybe a music
expert can comment on that. Anyway, if you
keep going on this circle of fifths, you'll eventually
fill in the chromatic notes, and arrive *almost* back
to the octave after a total of 12.
I believe that this is the method traditionally used,
with a slight modification, by piano tuners to get
an equal temperament scale.
The slight modification is that piano tuners
do not use exact fifths but count the beats
to get a very slightly different interval.
Martin Hogbin
.
|
|
|
|
| User: "Laurel Amberdine" |
|
| Title: Re: equal temperance vs. pure temperance music frequency ratios |
12 Sep 2003 11:53:31 PM |
|
|
On Fri, 12 Sep 2003 12:24:52 -0500, Eric Prebys <prebys@fnal.gov> wrote:
<large snip>
I have a brief description of this (with all the notes)
in Appendix C of the following:
http://viper.princeton.edu/~prebys/p104_s01/satlab/satlab2.html
which I wrote for a lab I was teaching once.
A web page! How did I not notice that you have a webpage? Two webpages
even! Very nice. :)
-Laurel
.
|
|
|
|
| User: "Lucius Chiaraviglio" |
|
| Title: Re: equal temperance vs. pure temperance music frequency ratios |
01 Oct 2003 04:11:14 PM |
|
|
On Fri, 12 Sep 2003 12:24:52 -0500, Eric Prebys wrote (in sci.physics
only):
[. . .]
If you start at a particular note, you can generate a untempered major
scale by making a "circle of fifths" (i.e. going up 3/2 in frequency)
and then dropping back (dividing by 2) when necessary to stay in a
single octave. Interestingly, if you
start on A, what you'll get is an E-major scale. I think that long ago
they made the choice to change it because the fourth is weird otherwise,
but maybe a music expert can comment on that. Anyway, if you keep going
on this circle of fifths, you'll eventually fill in the chromatic notes,
and arrive *almost* back to the octave after a total of 12.
If you do this enough times to make most of 4 circles of fifths, you
find that at note 42, the frequency of note 42 (16,585,998.48141 times the
nearest power of 2) only differs from the nearest power of 2 (16,777,216 =
2^24) by -191218 (1.14% difference) times the original note. So you don't
get exactly back to where you started, but you get so close that if you
spread the difference around the 41-note super-circle you just made, the
error is only -0.027%, much less than the -0.254% error that you have to
spread around to make a 12-note equal temperament.
To see what I mean, paste the following lines into Gnumeric or Excel:
1 =ln(A1)/ln(2) =round(B1) =2^C1 =A1-D1 =E1/D1
=A1*1.5 =ln(A2)/ln(2) =round(B2) =2^C2 =A2-D2 =E2/D2
and then replicate the second line (allowing your spreadsheet to increment
the line numbers normally) down to at least line 42. At G3 tack on:
=2^(1/6) =(G3-F3)-1
and at H42 tack on:
=F42/42
and then compare H3 to H42.
Did anyone ever try to make a 41-note microtonal system? (I seem to
recall someone in the Renaissance making a harpsichord with 30-odd keys
per octave, but I can't remember one with 41 keys per octave.)
--
Lucius Chiaraviglio
Approximate E-mail address:
To get the exact address: ^^^ ^replace this with 'r'
|||
replace this with single digit meaning the same thing
(Spambots of Doom, take that!).
.
|
|
|
|
|
