{"id":625,"date":"2021-01-14T17:09:02","date_gmt":"2021-01-14T16:09:02","guid":{"rendered":"http:\/\/www.jacques-rougemont.ch\/?p=625"},"modified":"2021-01-15T09:45:27","modified_gmt":"2021-01-15T08:45:27","slug":"the-harmonic-series","status":"publish","type":"post","link":"https:\/\/www.jacques-rougemont.ch\/?p=625&lang=en","title":{"rendered":"The harmonic series"},"content":{"rendered":"\n<p>A sound wave of well-defined pitch is by definition a periodic function of time $t$ with a corresponding spectrum $\\{\\alpha_n\\,:\\,n=1,2,\\dots\\}$ based on a fundamental frequency $\\omega$:<br>$$\\phi_\\omega(t)\\,=\\,\\alpha_0+\\sum_{n=1,2,\\dots}\\alpha_n\\cos\\bigl(2\\pi n\\omega (t-\\tau_n)\\bigr)~.$$<br>The set of frequencies which appear in this formula is called the harmonic series: frequencies ${\\omega, 2\\omega, 3\\omega, 4\\omega, \\dots}$ are respectively called the fundamental ($n=1$), first harmonic ($n=2$), second harmonic ($n=3$), etc. We can think of each component of the wave as an additional note above the fundamental as in Figure 1.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" width=\"785\" height=\"125\" src=\"http:\/\/www.jacques-rougemont.ch\/wp-content\/uploads\/2020\/05\/HarmonicSeries.png\" alt=\"\" class=\"wp-image-384\" srcset=\"https:\/\/www.jacques-rougemont.ch\/wp-content\/uploads\/2020\/05\/HarmonicSeries.png 785w, https:\/\/www.jacques-rougemont.ch\/wp-content\/uploads\/2020\/05\/HarmonicSeries-300x48.png 300w, https:\/\/www.jacques-rougemont.ch\/wp-content\/uploads\/2020\/05\/HarmonicSeries-768x122.png 768w, https:\/\/www.jacques-rougemont.ch\/wp-content\/uploads\/2020\/05\/HarmonicSeries-624x99.png 624w\" sizes=\"(max-width: 785px) 100vw, 785px\" \/><figcaption>The harmonic series of <strong>C<\/strong>. Numbers above the staff are <a href=\"http:\/\/www.jacques-rougemont.ch\/?p=194#pureInts\">deviations with respect to the actual tempered note<\/a>.<\/figcaption><\/figure>\n\n\n\n<p>The frequency $\\omega$ gives its name to the resulting wave, for example any wave with fundamental frequency $\\omega=440$ [Hz] is an <strong>A<\/strong> irrespective of which other coefficients appear in the series. When fundamentals of two notes are in a ratio of $2$ ($\\omega_2=2\\cdot\\omega_1$) , they form an octave and share the same name. Therefore notes with $\\omega = 110, 220, 440, 880$ [Hz] are all called <strong>A<\/strong>. Notice that the harmonic series of <strong>A<\/strong>$=880$ [Hz] is entirely contained within the harmonic series of <strong>A<\/strong>$=440$ [Hz] and similarly for any harmonic of any other note in the series (for example harmonics of <strong>E<\/strong>$=1320$ [Hz] or <strong>C\u266f<\/strong>$=2200$ [Hz] are all harmonics of <strong>A<\/strong>$=440$ [Hz]). These notes would however be considered &#8220;out of tune&#8221; in a perfectly equal temperament (which imposes <strong>E<\/strong>$=1318.51$ [Hz] and <strong>C\u266f<\/strong>$=2217.46$ [Hz]).<\/p>\n","protected":false},"excerpt":{"rendered":"<p>A sound wave of well-defined pitch is by definition a periodic function of time $t$ with a corresponding spectrum $\\{\\alpha_n\\,:\\,n=1,2,\\dots\\}$ based on a fundamental frequency $\\omega$:$$\\phi_\\omega(t)\\,=\\,\\alpha_0+\\sum_{n=1,2,\\dots}\\alpha_n\\cos\\bigl(2\\pi n\\omega (t-\\tau_n)\\bigr)~.$$The set of frequencies which appear in this formula is called the harmonic series: frequencies ${\\omega, 2\\omega, 3\\omega, 4\\omega, \\dots}$ are respectively called the fundamental ($n=1$), first harmonic [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":[],"categories":[20,22,26],"tags":[],"_links":{"self":[{"href":"https:\/\/www.jacques-rougemont.ch\/index.php?rest_route=\/wp\/v2\/posts\/625"}],"collection":[{"href":"https:\/\/www.jacques-rougemont.ch\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.jacques-rougemont.ch\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.jacques-rougemont.ch\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.jacques-rougemont.ch\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=625"}],"version-history":[{"count":4,"href":"https:\/\/www.jacques-rougemont.ch\/index.php?rest_route=\/wp\/v2\/posts\/625\/revisions"}],"predecessor-version":[{"id":632,"href":"https:\/\/www.jacques-rougemont.ch\/index.php?rest_route=\/wp\/v2\/posts\/625\/revisions\/632"}],"wp:attachment":[{"href":"https:\/\/www.jacques-rougemont.ch\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=625"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.jacques-rougemont.ch\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=625"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.jacques-rougemont.ch\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=625"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}