User:Hkm/Intro page: Difference between revisions

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== Why have I heard claims that our dominant tuning system is out of tune? ==

just intonation

=== just intonation ===

==== Through matching partials ====

If an instrument generates a single note, when air pressure at a given point near the instrument is graphed, the resulting graph is a sum of sine waves of various levels of frequency and amplitude. If the instrument has a harmonic timbre (as is the case with most methods of sound production, including the human voice, most instruments, and square/saw/triangle waves), these sine waves will have frequencies that are multiples of the fundamental tone (or the frequency of the perceived pitch).

When two notes are played at once where their frequency spectra share a high-amplitude frequency, these two notes sound concordant when played together. If these notes have a harmonic timbre, this occurs when the frequencies of the two notes have frequencies a and b where b/a is rational (i. e. the interval is within just intonation) and a and b are not too large. The two frequency spectra then share frequencies at [[Lcm|LCM]](a, b) and all multiples of it.

For example, if we take two frequencies at 100 Hz (called "Do" here) and 150 Hz ("Sol"), we see that the notes share frequencies at LCM(100, 150) = 300 and every multiple of it.<pre>Tone Frequencies of partials (Hz)

Do 100 200 300 400 500 600 700 800 900 ...

So 150 300 450 600 750 900 1050 1200 1350 ...</pre>These partials will "meld" when our Do and Sol are played together. This goes by the wonderful name of ''Tonverschmelzung'' in German. It is a very distinctive "blending" sound. If our Sol was tuned to, for example, 148 Hz, its second harmonic component would be at 296 Hz, and the two tones played together would not meld together at 300 Hz, but would "beat". That is, we would hear a throbbing sound, the "beat rate" of which is found by reckoning the distance in Hertz between the two near-coincident partials. In this case, 300 - 296 = 4 Hz, so we'd hear a beating of four times a second (this is like a rhythm of eighth notes at a metronome marking of 120 beats per minute). This melding and beating makes it easy to sing JI intervals.

===Through other phenomena===

this needs further explanation

edos

@@ Line 8: / Line 8: @@
 == Why have I heard claims that our dominant tuning system is out of tune? ==
-just intonation
+=== just intonation ===
+==== Through matching partials ====
+If an instrument generates a single note, when air pressure at a given point near the instrument is graphed, the resulting graph is a sum of sine waves of various levels of frequency and amplitude. If the instrument has a harmonic timbre (as is the case with most methods of sound production, including the human voice, most instruments, and square/saw/triangle waves), these sine waves will have frequencies that are multiples of the fundamental tone (or the frequency of the perceived pitch).
+When two notes are played at once where their frequency spectra share a high-amplitude frequency, these two notes sound concordant when played together. If these notes have a harmonic timbre, this occurs when the frequencies of the two notes have frequencies a and b where b/a is rational (i. e. the interval is within just intonation) and a and b are not too large. The two frequency spectra then share frequencies at [[Lcm|LCM]](a, b) and all multiples of it.
+For example, if we take two frequencies at 100 Hz (called "Do" here) and 150 Hz ("Sol"), we see that the notes share frequencies at LCM(100, 150) = 300 and every multiple of it.<pre>Tone   Frequencies of partials (Hz)
+Do     100  200  300  400  500  600  700  800  900  ...
+So     150  300  450  600  750  900 1050 1200 1350  ...</pre>These partials will "meld" when our Do and Sol are played together. This goes by the wonderful name of ''Tonverschmelzung'' in German. It is a very distinctive "blending" sound. If our Sol was tuned to, for example, 148 Hz, its second harmonic component would be at 296 Hz, and the two tones played together would not meld together at 300 Hz, but would "beat". That is, we would hear a throbbing sound, the "beat rate" of which is found by reckoning the distance in Hertz between the two near-coincident partials. In this case, 300 - 296 = 4 Hz, so we'd hear a beating of four times a second (this is like a rhythm of eighth notes at a metronome marking of 120 beats per minute). This melding and beating makes it easy to sing JI intervals.
+===Through other phenomena===
+this needs further explanation
 edos