Xen concepts for beginners: Difference between revisions

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== Interval math ==

Xen discussion uses ~~both~~ *frequency ratios* ~~and~~ *logarithmic units* such as cents and edo steps~~. To stack two intervals, we use different types of operations for~~ the two kinds of units. To stack two intervals written as ratios, we *multiply*, whereas to stack two intervals written as cents or edo steps, we *add* the intuitive way. To "unstack" an interval from another interval, we *divide* the respective ratios and *subtract* logarithmic units. To convert between cents and ratios we use the following formulas:

Xen discussion uses two kinds of units:

* ''frequency ratios''

** the frequency is the absolute pitch of any given tone usually in hertz (Hz). The ratio between frequencies is just a number.

* ''logarithmic units'' such as cents and edo steps that give the offset from two

`cents = 1200*log(ratio)/log(2)`

To stack two intervals, we use different types of operations for the two kinds of units. To stack two intervals written as ratios, we *multiply*, whereas to stack two intervals written as cents or edo steps, we *add* the intuitive way. To "unstack" an interval from another interval, we *divide* the respective ratios and *subtract* logarithmic units. To convert between cents and ratios we use the following formulas:

`ratio = 2^(cents/1200)`

cents = 1200*log(ratio)/log(2)

ratio = 2^(cents/1200)

The unison has frequency ratio 1/1 and is 0 cents. The octave has frequency ratio 2/1 and is exactly 1200 cents. A 12edo/12tet semitone has frequency ratio 2^(1/12) and is exactly 100 cents.

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The notation m\n means m steps of n-edo. 7\12 is 7 steps out of 12edo, the 12edo perfect fifth.

A very important operation in xen math is the mediant. The mediant of two fractions, a/b and c/d, is the "freshman sum" (a+b)/(c+d). For example, the mediant of 4/3, the just perfect fourth, and 5/4, the just major third, is 9/7, the supermajor third. If two fractions are in lowest terms, their mediant is the simplest fraction that is strictly between both. The mediant is commonly used for both JI ratios and edo intervals.

A very important operation in xen math is the [[mediant]]. The mediant of two fractions, a/b and c/d, is the "freshman sum" (a+b)/(c+d). For example, the mediant of 4/3, the just perfect fourth, and 5/4, the just major third, is 9/7, the supermajor third. If two fractions are in lowest terms, their mediant is the simplest fraction that is strictly between both. The mediant is commonly used for both JI ratios and edo intervals.

Another important operation is reduction. To reduce an interval a by an interval b means to stack or "unstack" b from a until a is at least the unison and less than b. For example, 3/1 reduced by 2/1 is 3/2.

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 == Interval math ==
-Xen discussion uses both *frequency ratios* and *logarithmic units* such as cents and edo steps. To stack two intervals, we use different types of operations for the two kinds of units.  To stack two intervals written as ratios, we *multiply*, whereas to stack two intervals written as cents or edo steps, we *add* the intuitive way. To "unstack" an interval from another interval, we *divide* the respective ratios and *subtract* logarithmic units. To convert between cents and ratios we use the following formulas:
+Xen discussion uses two kinds of units:
+* ''frequency ratios''
+** the frequency is the absolute pitch of any given tone usually in hertz (Hz). The ratio between frequencies is just a number.
+* ''logarithmic units'' such as cents and edo steps that give the offset from two
-`cents = 1200*log(ratio)/log(2)`
+To stack two intervals, we use different types of operations for the two kinds of units.  To stack two intervals written as ratios, we *multiply*, whereas to stack two intervals written as cents or edo steps, we *add* the intuitive way. To "unstack" an interval from another interval, we *divide* the respective ratios and *subtract* logarithmic units. To convert between cents and ratios we use the following formulas:
-`ratio = 2^(cents/1200)`
+ cents = 1200*log(ratio)/log(2)
+ ratio = 2^(cents/1200)
 The unison has frequency ratio 1/1 and is 0 cents. The octave has frequency ratio 2/1 and is exactly 1200 cents. A 12edo/12tet semitone has frequency ratio 2^(1/12) and is exactly 100 cents.
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 The notation m\n means m steps of n-edo. 7\12 is 7 steps out of 12edo, the 12edo perfect fifth.
-A very important operation in xen math is the mediant. The mediant of two fractions, a/b and c/d, is the "freshman sum" (a+b)/(c+d). For example, the mediant of 4/3, the just perfect fourth, and 5/4, the just major third, is 9/7, the supermajor third. If two fractions are in lowest terms, their mediant is the simplest fraction that is strictly between both. The mediant is commonly used for both JI ratios and edo intervals.
+A very important operation in xen math is the [[mediant]]. The mediant of two fractions, a/b and c/d, is the "freshman sum" (a+b)/(c+d). For example, the mediant of 4/3, the just perfect fourth, and 5/4, the just major third, is 9/7, the supermajor third. If two fractions are in lowest terms, their mediant is the simplest fraction that is strictly between both. The mediant is commonly used for both JI ratios and edo intervals.
 Another important operation is reduction. To reduce an interval a by an interval b means to stack or "unstack" b from a until a is at least the unison and less than b. For example, 3/1 reduced by 2/1 is 3/2.