Melodic irrationalities - Cronologia

Renzo il 12:33, 11 mag 2014

2014-05-11T12:33:02Z

Renzo il 12:30, 11 mag 2014

2014-05-11T12:30:26Z

Renzo il 12:28, 11 mag 2014

2014-05-11T12:28:14Z

Renzo il 06:51, 10 mag 2014

2014-05-10T06:51:26Z

Renzo il 16:14, 9 mag 2014

2014-05-09T16:14:53Z

Renzo il 16:07, 9 mag 2014

2014-05-09T16:07:47Z

Renzo: Creata pagina con '==Melodic irrationalities== Melodic irrationalities. The tempered scale is a way to equally distribute the errors between notes. When an instrument is tuned according the te...'

2014-05-09T16:07:16Z

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==Melodic irrationalities==

Melodic irrationalities.

The tempered scale is a way to equally distribute the errors between
notes. When an instrument is tuned according the tempered scale it
means that it has been put out of tune in an homogeneous manner.

But I have always wondered about the idea of harmony. Our tempered
chords are less or more melodic than their perfect counterparts?
I mean: are we loosing something because our instruments have to
approximate rational numbers with the closest step in a discretization
of a equispaced sequence of frequency logarithms?

A perfect fifth (in mathematical meaning) is the ratio between
the third and second harmonic: 3/2.
A perfect third is the ratio between the fifth and fourth: 5/4.
A "perfect" A major chord is then composed by the following three
frequencies in hz:
[440, 550, 660]
but, due to the tempered tuning, when we play that chord on an instrument
(say a keyboard) we force our tool to emit the following frequencies:
[440, 554.3653..., 659.2551...]
i.e. A=440hz, A times 2^(5/12), A times 2^(7/12)
Similarly the perfect minor chord is:
[440, 528, 660]
as the natural third has the ratio 6/5 and the tempered minor chord is:
[440, 523.2511..., 659.2551...]

Lissajous curves appear on a oscilloscope when the tracing point is driven
in X/Y mode by two sinusoidal inputs having a rational ratio between their
frequency.
In order to plot a chord we'd need a 3D oscilloscope, which does not exist
as a physical tool (as far as I know).
In fact, the three notes of the chord has to drive the position along the
three axis of our virtual oscilloscope.

I implemented a (quick and dirty) 3D virtual oscilloscope using Python and
and the vpython library.

The result is documented in the following movies.
[http://www.cs.unibo.it/~renzo/music/out.still.mpg First Movie]
[http://www.cs.unibo.it/~renzo/music/out.3D.mpg Second Movie]
In each movie the sequence of the chords is: perfect Major, tempered Major,
perfect minor, tempered minor. The soudtrack is always consistent with
the plotted chord.
In the first movie we can see one still projection of each 3D curve.
As the reader can see, the perfect chords are 3D Lissajous curves. The minor
chord is more convoluted than the major chord, both seem to be strange knots.
The signal redraws these "knots" passing all the times on the same
path, generating the color effects on the "ropes".
On the contrary in the tempered scale the frequency ratios are irrational
numbers, so the path never returns on its previous trajectory.

In the second movie, I have changed the perspective, the pont of vision
during the animation, to that the viewer can have a view of the chords
on different perspectives.

The difference in the sound is subtle, but can be perceived.
To me, the perfect chords seem more flat, maybe too perfect to express human
feelings. The tempered chords, with their imperfections, sound
better. This continuous evolution of the correlation among the three
tones of the chord can create other effects, beats.
On some projetions of the 3D animation, the tempered chord generate some
temporary fish-shaped trajectories which appear, disappear, appear again
but the tail of the fish is on the other direction.

Is my preference related to my culture? I am used to the tempered 12 semitones
scale as I have grown up in the twentieth (and twentyfirst) century and
I am living in the west world.

Anyway I like the idea to be able to watch the chords, and to watch their
appearance. If I met them in person, I can say that I know them better, now.

(C) CC-BY-SA Unported Renzo Davoli 2014

By the virtual 3D oscilloscope I can observe all the 3D curves generated by the
four chords above and I can compare them with the sound of the chords.

@@ Riga 76: / Riga 76: @@
 In [http://www.cs.unibo.it/~renzo/music this directory] you'll find:
-* toneM.wav toneN.wav: the major chord, thr former is the tempered one, the latter is the mathematical chord (or natural
+* toneM.wav toneN.wav: the major chords, the former is the tempered one, the latter is the mathematical chord (or natural
 or perfect)
-* tonem.wav tonen.wav: the minor chord, in the same way tonem is tempered, tonen is mathematical
+* tonem.wav tonen.wav: the minor chords, in the same way tonem is tempered, tonen is mathematical
-* chord2.py (python3 source, it needs the wave package). This program generated sound files of the chord you want to test. Use it in this way:
+* chord2.py (python3 source, it needs the wave package). This program generated sound files of the chord you want to test. Use it in this way (args are the three frequencies in Hz, the number of samples and the name of the output file):
     ./chords2.py 440 550 660 100000 /tmp/toneM.wav
     ./chords2.py 440.0 440.0*(2.0**(4.0/12)) 440.0*(2.0**(7.0/12)) 100000 /tmp/toneN.wav

← Versione meno recente		Versione delle 06:51, 10 mag 2014
Riga 72:		Riga 72:

	(C) CC-BY-SA Unported Renzo Davoli 2014		(C) CC-BY-SA Unported Renzo Davoli 2014
		+
		+	In [http://www.cs.unibo.it/~renzo/music this directory] you'll find:
		+	* toneM.wav toneN.wav: the major chord, thr former is the tempered one, the latter is the mathematical chord (or natural
		+	or perfect)
		+	* tonem.wav tonen.wav: the minor chord, in the same way tonem is tempered, tonen is mathematical
		+	* chord2.py (python3 source, it needs the wave package). This program generated sound files of the chord you want to test. Use it in this way:
		+	./chords2.py 440 550 660 100000 /tmp/toneM.wav
		+	./chords2.py 440.0 440.0(2.0(4.0/12)) 440.0(2.0**(7.0/12)) 100000 /tmp/toneN.wav
		+	* lissajous2.py (python2, needs python-visual a.k.a. vpython) gives you an interactive 3D view of the chord.
		+	./lissajous2.py 440 550 660
		+	./lissajous2.py 440.0 440.0(2.0(4.0/12)) 440.0(2.0**(7.0/12))
		+
		+	Using the resources here above you can repeat my experiments, watch the chords from different perspectives, create new experiments and/or listen carefully to the chord samples to test the difference between them.

@@ Riga 13: / Riga 13: @@
 A perfect third is the ratio between the fifth and fourth: 5/4.
 A "perfect" A major chord is then composed by the following three
-frequencies in hz:
+frequencies in Hz:
    [440, 550, 660]
 but, due to the tempered tuning, when we play that chord on an instrument
 (say a keyboard) we force our tool to emit the following frequencies:
    [440, 554.3653..., 659.2551...]
-i.e. A=440hz, A times 2^(5/12), A times 2^(7/12)
+i.e. A=440hz, A times 2^(5/12), A times 2^(7/12).
-Similarly the perfect minor chord is:
+Similarly, the perfect minor chord is:
    [440, 528, 660]
 as the natural third has the ratio 6/5 and the tempered minor chord is:

@@ Riga 38: / Riga 38: @@
 The result is documented in the following movies.
 [http://www.cs.unibo.it/~renzo/music/out.still.mpg First Movie]
 [http://www.cs.unibo.it/~renzo/music/out.3D.mpg Second Movie]
 In each movie the sequence of the chords is: perfect Major, tempered Major,
 perfect minor, tempered minor.  The soudtrack is always (almost) consistent with

@@ Riga 34: / Riga 34: @@
 I implemented a (quick and dirty) 3D virtual oscilloscope using Python and
 and the vpython library.
 The result is documented in the following movies.
@@ Riga 39: / Riga 41: @@
 [http://www.cs.unibo.it/~renzo/music/out.3D.mpg Second Movie]
 In each movie the sequence of the chords is: perfect Major, tempered Major,
-perfect minor, tempered minor.  The soudtrack is always consistent with
+perfect minor, tempered minor.  The soudtrack is always (almost) consistent with
 the plotted chord.
 In the first movie we can see one still projection of each 3D curve.
@@ Riga 49: / Riga 51: @@
 numbers, so the path never returns on its previous trajectory.
-In the second movie, I have changed the perspective, the point of vision
+In the second movie, I have changed the point of vision
 during the animation, to that the viewer can have a view of the chords
 on different perspectives.
@@ Riga 63: / Riga 65: @@
 Is my preference related to my culture? I am used to the tempered 12 semitones
-scale as I have grown up in the twentieth (and twentyfirst) century and
+scale as I grew up in the twentieth century, I am living in the twentyfirst and
-I am living in the west world.
+in the west world.
 Anyway I like the idea to be able to watch the chords, and to watch their
@@ Riga 70: / Riga 72: @@
 (C) CC-BY-SA Unported Renzo Davoli 2014

@@ Riga 1: / Riga 1: @@
 The tempered scale is a way to equally distribute the errors between
 notes. When an instrument is tuned according the tempered scale it
@@ Riga 53: / Riga 49: @@
 numbers, so the path never returns on its previous trajectory.
-In the second movie, I have changed the perspective, the pont of vision
+In the second movie, I have changed the perspective, the point of vision
 during the animation, to that the viewer can have a view of the chords
 on different perspectives.