meanqratapprox

From: Petr Pařízek (2005-12-17)
Subject: An excellent way of approximating quarter-comma meantone (just by ear)

Hi all.
About half a year ago, I was examining various ways of approximating
quarter-comma meantone just by ear. The method I finally chose is something
which I believe people must have noticed many centuries ago as it works
pretty nicely indeed. Applying this to the modern 21st century
possibilities, you may even take an ordinary handworn watch that makes
regular ticks usually at a speed of one per second (some watches tick 5
times per second, which is not very appropriate in these cases) as it is not
always possible to have a metronome with you. Of course, you can freely use
this method with any starting pitch you like. But the very best results come
out if you choose an A4 of 430Hz (i.e. as in the 18th century or so).
OK, now how to do it? As I wish the tuning to be a very good approximation
of quarter-comma meantone indeed, I start by tuning A4 to 430Hz. This is the
only situation when you need something like an external tuner as you
probably won't find the exact 430Hz just by ear. But from now on, you can
put your tuner away and just listen carefully. Now, I take D4 as a fifth
downwards from A4 and I tune it slightly higher than in the case of a pure
fifth - by "slightly higher" I mean in such a way that I can hear exactly 8
beats per 3 seconds. Then, I take E4 as a fourth downwards from A4 and I
tune it slightly lower than if the fourth was pure - by "slightly lower" I
mean in such a way that I can hear exactly 4 beats per second. Then, I take
B4 as a fifth upwards from E4 and I tune it slightly lower than for a pure
fifth - i.e. in such a way that I can hear exactly 3 beats per second.
Finally, the most time-consuming step takes place as all the other tones of
the meantone chain are tuned as pure major thirds or minor sixths from the
tones which I have already tuned.
Provided that it makes no harm to the instrument in question to have A4 at
430Hz and that the overtones of the instrument deviate from the harmonic
series no more than just very slightly, what comes out as a result is right
what I'm sending in the regular Scala format at the end of this message. If
you look at the chain carefully, you can see the maximum deviation from the
exact quarter-comma temperament is no more than about 1/60 of a cent!
Petr

! meanqratapprox.scl
! June 2005 - Petr Parizek
A very close approximation of quarter-comma meantone
 12
!
 5375/5144
 6470/5787
 3846/3215
 5/4
 860/643
 16175/11574
 1923/1286
 25/16
 1075/643
 10352/5787
 9615/5144
 2/1

From: Petr Pařízek (2005-12-17)
Subject: An excellent way of approximating quarter-comma meantone (just by ear)

Hi all.
About half a year ago, I was examining various ways of approximating
quarter-comma meantone just by ear. The method I finally chose is something
which I believe people must have noticed many centuries ago as it works
pretty nicely indeed. Applying this to the modern 21st century
possibilities, you may even take an ordinary handworn watch that makes
regular ticks usually at a speed of one per second (some watches tick 5
times per second, which is not very appropriate in these cases) as it is not
always possible to have a metronome with you. Of course, you can freely use
this method with any starting pitch you like. But the very best results come
out if you choose an A4 of 430Hz (i.e. as in the 18th century or so).
OK, now how to do it? As I wish the tuning to be a very good approximation
of quarter-comma meantone indeed, I start by tuning A4 to 430Hz. This is the
only situation when you need something like an external tuner as you
probably won't find the exact 430Hz just by ear. But from now on, you can
put your tuner away and just listen carefully. Now, I take D4 as a fifth
downwards from A4 and I tune it slightly higher than in the case of a pure
fifth - by "slightly higher" I mean in such a way that I can hear exactly 8
beats per 3 seconds. Then, I take E4 as a fourth downwards from A4 and I
tune it slightly lower than if the fourth was pure - by "slightly lower" I
mean in such a way that I can hear exactly 4 beats per second. Then, I take
B4 as a fifth upwards from E4 and I tune it slightly lower than for a pure
fifth - i.e. in such a way that I can hear exactly 3 beats per second.
Finally, the most time-consuming step takes place as all the other tones of
the meantone chain are tuned as pure major thirds or minor sixths from the
tones which I have already tuned.
Provided that it makes no harm to the instrument in question to have A4 at
430Hz and that the overtones of the instrument deviate from the harmonic
series no more than just very slightly, what comes out as a result is right
what I'm sending in the regular Scala format at the end of this message. If
you look at the chain carefully, you can see the maximum deviation from the
exact quarter-comma temperament is no more than about 1/60 of a cent!
Petr

! meanqratapprox.scl
! June 2005 - Petr Parizek
A very close approximation of quarter-comma meantone
 12
!
 5375/5144
 6470/5787
 3846/3215
 5/4
 860/643
 16175/11574
 1923/1286
 25/16
 1075/643
 10352/5787
 9615/5144
 2/1

From: Petr Pařízek (2005-12-17)
Subject: Re: [tuning] An excellent way of approximating quarter-comma meantone (just by ear)

I forgot to add "(1/1 = 257.2Hz)" at the end of the description line.