Editing Musical acoustics (section)

==Harmony==

{{Main|Harmony}}

If two notes are simultaneously played, with frequency [[ratio]]s that are simple fractions (e.g. {{small| {{sfrac| 2 | 1 }} }}, {{small| {{sfrac| 3 | 2 }} }}, or {{small| {{sfrac| 5 | 4 }} }}), the composite wave is still periodic, with a short period – and the combination sounds [[consonance and dissonance|consonant]]. For instance, a note vibrating at 200&nbsp;[[Hertz (unit)|Hz]] and a note vibrating at 300&nbsp;Hz (a [[perfect fifth]], or {{nobr|{{small| {{sfrac| 3 | 2 }} }} ratio,}} above 200&nbsp;Hz) add together to make a wave that repeats at 100&nbsp;Hz: Every {{small| {{sfrac| 1 | 100 }} }} of a second, the 300&nbsp;Hz wave repeats three times and the 200&nbsp;Hz wave repeats twice. Note that the ''combined'' wave repeats at 100&nbsp;Hz, even though there is no actual 100&nbsp;Hz sinusoidal component contributed by an individual sound source.

Additionally, the two notes from acoustical instruments will have overtone partials that will include many that share the same frequency. For instance, a note with the frequency of its fundamental harmonic at 200&nbsp;[[Hertz (unit)|Hz]] can have harmonic overtones at: 400, 600, 800, {{gaps|1|000}}, {{gaps|1|200}}, {{gaps|1|400}}, {{gaps|1|600}}, {{gaps|1|800}}, ...&nbsp;[[Hertz (unit)|Hz]].
A note with fundamental frequency of 300&nbsp;Hz can have overtones at: 600, 900, {{gaps|1|200}}, {{gaps|1|500}}, {{gaps|1|800}}, ...&nbsp;[[Hertz (unit)|Hz]].
The two notes share harmonics at 600, {{gaps|1|200}}, {{nobr|{{gaps|1|800}} Hz,}} and more that coincide with each other, further along in the each series.

Although the mechanism of human hearing that accomplishes it is still incompletely understood, practical musical observations for nearly {{nobr|{{gaps|2|000}} years}}<ref>{{cite book |first=Gaius Claudius |last=Ptolemy |author-link=Claudius Ptolemy |id=year {{nobr|{{circa|180 {{sc|ce}}}}}} |trans-title=Harmonics |script-title=he:{{math|Ἁρμονικόν}} |title=Harmonikon  |title-link=Harmonics}}</ref> The combination of composite waves with short fundamental frequencies and shared or closely related partials is what causes the sensation of harmony: When two frequencies are near to a simple fraction, but not exact, the composite wave cycles slowly enough to hear the cancellation of the waves as a steady pulsing instead of a tone. This is called [[beat (acoustics)|beating]], and is considered unpleasant, or [[consonance and dissonance|dissonant]].

The frequency of beating is calculated as the difference between the frequencies of the two notes. When two notes are close in pitch they beat slowly enough that a human can measure the frequency ''difference'' by ear, with a [[stopwatch]]; beat timing is how tuning pianos, harps, and [[harpsichord]]s to complicated [[temperament (music)|temperaments]] was managed before affordable [[electronic tuner|tuning meters]].
* For the example above, {{nobr|{{math| {{big|'''&#x7C;'''}} 200 }}[[Hertz (unit)|Hz]]{{math| − 300 }}Hz{{math| {{big|'''&#x7C;'''}} &thinsp;{{=}}&thinsp; 100}} Hz .}}

* As another example from [[modulation]] theory, a combination of {{nobr|&thinsp;{{gaps|3|425}} Hz&thinsp;}} and {{nobr|&thinsp;{{gaps|3|426}} Hz&thinsp;}} would beat once per second, since {{nobr| &thinsp;{{math| {{big|'''&#x7C;'''}} {{gaps|3|425}} }}Hz{{math| − {{gaps|3|426}} }}Hz{{math| {{big|'''&#x7C;'''}} &thinsp;{{=}}&thinsp; 1}} Hz .}}

The difference between consonance and dissonance is not clearly defined, but the higher the beat frequency, the more likely the interval is dissonant. [[Hermann von Helmholtz|Helmholtz]] proposed that maximum dissonance would arise between two pure tones when the beat rate is roughly 35&nbsp;Hz.<ref>{{cite web |title=Roughness |series=Music&nbsp;829B |type=course notes |website=music-cog.ohio-state.edu |publisher=[[Ohio State University]] |url=http://www.music-cog.ohio-state.edu/Music829B/roughness.html }}</ref>