User:VectorGraphics/Intro to xenharmony: Difference between revisions

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== 2. L + Ratio ==

Because pitch perception is logarithmic, any interval can be characterized as a ratio between pitches. The easiest such ratios to tune are simple ratios of whole numbers: 2/1, 3/2, 4/3, 5/4, 5/3, etc, so many historical instruments were tuned to these intervals. Using whole number ratios also has another advantage in that due to the fundamental theorem of arithmetic, which says that any rational number can be written in terms of powers of primes in exactly one way, there is an inherent mathematical structure to any set of whole number ratios in use. Additionally, whole number ratios are easy to recognize by ear, as they produce an auditory effect called "concordance", associated with, on certain timbres, a buzzing effect. Concordance is, essentially, how much two notes feel like "the same note": think about notes an octave up or down in standard music theory.

Because pitch perception is logarithmic, any interval can be characterized as a ratio (or more generally, a multiplicative relationship, which can be an irrational number) between pitches. The easiest such ratios to tune are simple ratios of whole numbers: 2/1, 3/2, 4/3, 5/4, 5/3, etc, so many historical instruments were tuned to these intervals. Using whole number ratios also has another advantage in that due to the fundamental theorem of arithmetic, which says that any rational number can be written in terms of powers of primes in exactly one way, there is an inherent mathematical structure to any set of whole number ratios in use. Additionally, whole number ratios are easy to recognize by ear, as they produce an auditory effect called "concordance", associated with, on certain timbres, a buzzing effect. Concordance is, essentially, how much two notes feel like "the same note": think about notes an octave up or down in standard music theory.

== 3. Making Steps ==

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 == 2. L + Ratio ==
-Because pitch perception is logarithmic, any interval can be characterized as a ratio between pitches. The easiest such ratios to tune are simple ratios of whole numbers: 2/1, 3/2, 4/3, 5/4, 5/3, etc, so many historical instruments were tuned to these intervals. Using whole number ratios also has another advantage in that due to the fundamental theorem of arithmetic, which says that any rational number can be written in terms of powers of primes in exactly one way, there is an inherent mathematical structure to any set of whole number ratios in use. Additionally, whole number ratios are easy to recognize by ear, as they produce an auditory effect called "concordance", associated with, on certain timbres, a buzzing effect. Concordance is, essentially, how much two notes feel like "the same note": think about notes an octave up or down in standard music theory.
+Because pitch perception is logarithmic, any interval can be characterized as a ratio (or more generally, a multiplicative relationship, which can be an irrational number) between pitches. The easiest such ratios to tune are simple ratios of whole numbers: 2/1, 3/2, 4/3, 5/4, 5/3, etc, so many historical instruments were tuned to these intervals. Using whole number ratios also has another advantage in that due to the fundamental theorem of arithmetic, which says that any rational number can be written in terms of powers of primes in exactly one way, there is an inherent mathematical structure to any set of whole number ratios in use. Additionally, whole number ratios are easy to recognize by ear, as they produce an auditory effect called "concordance", associated with, on certain timbres, a buzzing effect. Concordance is, essentially, how much two notes feel like "the same note": think about notes an octave up or down in standard music theory.
 == 3. Making Steps ==