<story><title>Why are D-sharp and E-flat considered to be two different notes?</title><url>https://www.ethanhein.com/wp/2022/why-are-d-sharp-and-e-flat-considered-to-be-two-different-notes/</url></story><parent_chain><item><author>evrydayhustling</author><text>Now that we can have electronic instruments that &amp;quot;tune&amp;quot; themselves, could we compute song-optimal tunings that preserve the intervals used most in that song?  Does this have a name?&lt;p&gt;As a guitarist we often swap guitars or retune to make certain songs easier to play, or to be able to get a certain tamber put of the note.  But I never considered it as a way to address temperament.&lt;p&gt;It&amp;#x27;s interesting to think how much of music theory emerges out of reconciliation with available instruments, as opposed to reconciliation with the ear.</text></item><item><author>perihelions</author><text>The paradox is that you can&amp;#x27;t create a theory of music whose notes are both (a) evenly spaced and (b) contain the integer ratios.&lt;p&gt;You want (a) because it you gives you nice algebraic properties (the music structure is invariant under frequency shifts). You want (b) because small-integer ratios are pleasant sounding -- partly culturally-acquired taste, partly because physics gives musical instruments acoustic spectra in integral multiples of a fundamental frequency: f, 2f, 3f ... nf. Small-integer ratios are naturally occurring and very recognizable.&lt;p&gt;Modern tuning (C-f &amp;quot;12-TET&amp;quot; in the article) almost, approximately satisfies (a) and (b) simultaneously. &amp;quot;12&amp;quot; means there&amp;#x27;s twelve tones between f and 2f; the ratio between adjacent tones is defined to be 2^{1&amp;#x2F;12}. This tuning can&amp;#x27;t contain both f and 3f (so it fails (b)), but it *can* contain f and 2^{19&amp;#x2F;12}f ~ 2.9966f, which is actually close enough to 3f to be indistinguishable. (Almost works!) But as you build ratios out of larger integers, it audibly falls apart. The closest you can get to (5&amp;#x2F;3)f is 2^{9&amp;#x2F;12}f = ~1.6818f, which is already 10% of the way to the next note. And it rapidly gets worse.&lt;p&gt;This is why two on-paper-identical notes can end up audibly different, depending on what key you&amp;#x27;re starting with (and hence how they are approached). There&amp;#x27;s tension internal to music theory itself.</text></item></parent_chain><comment><author>xhevahir</author><text>You might like to hear this proprietary algorithm: &lt;a href=&quot;http:&amp;#x2F;&amp;#x2F;www.hermode.com&amp;#x2F;index_en.html&quot; rel=&quot;nofollow&quot;&gt;http:&amp;#x2F;&amp;#x2F;www.hermode.com&amp;#x2F;index_en.html&lt;/a&gt;&lt;p&gt;Since you&amp;#x27;re a guitarist, there&amp;#x27;s also this Swedish guitar, which purports to solve the tuning problem (which I tend to think is not a problem but an essential part of the instrument&amp;#x27;s sound) &lt;a href=&quot;https:&amp;#x2F;&amp;#x2F;youtu.be&amp;#x2F;-penQWPHJzI&quot; rel=&quot;nofollow&quot;&gt;https:&amp;#x2F;&amp;#x2F;youtu.be&amp;#x2F;-penQWPHJzI&lt;/a&gt;</text></comment>
<story><title>Why are D-sharp and E-flat considered to be two different notes?</title><url>https://www.ethanhein.com/wp/2022/why-are-d-sharp-and-e-flat-considered-to-be-two-different-notes/</url></story><parent_chain><item><author>evrydayhustling</author><text>Now that we can have electronic instruments that &amp;quot;tune&amp;quot; themselves, could we compute song-optimal tunings that preserve the intervals used most in that song?  Does this have a name?&lt;p&gt;As a guitarist we often swap guitars or retune to make certain songs easier to play, or to be able to get a certain tamber put of the note.  But I never considered it as a way to address temperament.&lt;p&gt;It&amp;#x27;s interesting to think how much of music theory emerges out of reconciliation with available instruments, as opposed to reconciliation with the ear.</text></item><item><author>perihelions</author><text>The paradox is that you can&amp;#x27;t create a theory of music whose notes are both (a) evenly spaced and (b) contain the integer ratios.&lt;p&gt;You want (a) because it you gives you nice algebraic properties (the music structure is invariant under frequency shifts). You want (b) because small-integer ratios are pleasant sounding -- partly culturally-acquired taste, partly because physics gives musical instruments acoustic spectra in integral multiples of a fundamental frequency: f, 2f, 3f ... nf. Small-integer ratios are naturally occurring and very recognizable.&lt;p&gt;Modern tuning (C-f &amp;quot;12-TET&amp;quot; in the article) almost, approximately satisfies (a) and (b) simultaneously. &amp;quot;12&amp;quot; means there&amp;#x27;s twelve tones between f and 2f; the ratio between adjacent tones is defined to be 2^{1&amp;#x2F;12}. This tuning can&amp;#x27;t contain both f and 3f (so it fails (b)), but it *can* contain f and 2^{19&amp;#x2F;12}f ~ 2.9966f, which is actually close enough to 3f to be indistinguishable. (Almost works!) But as you build ratios out of larger integers, it audibly falls apart. The closest you can get to (5&amp;#x2F;3)f is 2^{9&amp;#x2F;12}f = ~1.6818f, which is already 10% of the way to the next note. And it rapidly gets worse.&lt;p&gt;This is why two on-paper-identical notes can end up audibly different, depending on what key you&amp;#x27;re starting with (and hence how they are approached). There&amp;#x27;s tension internal to music theory itself.</text></item></parent_chain><comment><author>ajross</author><text>&amp;gt; Now that we can have electronic instruments that &amp;quot;tune&amp;quot; themselves, could we compute song-optimal tunings that preserve the intervals used most in that song?&lt;p&gt;We&amp;#x27;ve had self-tuning instruments for thousands of years.  Vocal harmony has almost always been perfectly tuned for its key.  Likewise orchestral strings are fretless and can produce perfect intervals.  Equitemperment was an innovation in the 17th century because it approximated the perfect intervals very well (&amp;quot;sounded good&amp;quot;) but also permitted the ability to simultaneously represent scales based on every note in the circle of fifths (&amp;quot;sounded interesting&amp;quot;).  But the &amp;quot;real&amp;quot; chords were always (well, since the late middle ages) understood to be integer ratios.</text></comment>