Arithmetic tuning: Difference between revisions

An '''arithmetic tuning''' is one which has equal step sizes ''of any kind of quantity'', whether that be pitch, frequency, or length (of the resonating entity producing the sound).

 

This sense of the word arithmetic comes from the mathematical concept of an [[Wikipedia:Arithmetic sequence|arithmetic sequence]], which means a sequence with a constant difference between successive terms.  

== Types ==

* [[OD|OD, otonal division]] (such as an [[ODO]])

* [[UD|UD, utonal division]] (such as a [[UDO]])

== Basic examples ==

# the ''overtone'' series has equal steps of ''frequency'' (1, 2, 3, 4, etc.; adding 1 each step)

# any ''edo'' has equal steps of ''pitch'' (12edo goes 0\12, 1\12, 2\12, 3\12, etc.; adding 1\12 each step)

# the ''undertone'' series has equal steps of ''length'' (to play the first four steps of the undertone series you would pluck the whole length of a string, then 3/4 the string, then 2/4, then 1/4; adding -1/4 length each step)

== Sequences ==

Other arithmetic tunings can be found by changing the step size. For example, if you vary the overtone series to have a step size of 3/4 instead of 1, then you get the tuning <math>1, 1\frac 34, 2\frac 24, 3\frac14</math><span>, which is equivalent to <math>\frac 44, \frac 74, \frac{10}{4}, \frac{13}{4}</math>, or in other words, a class iii [[isoharmonic chords|isoharmonic]] tuning with starting position of 4. We call this the otonal sequence of 3 over 4, or OS3/4.  

If the new step size is irrational, the tuning is no longer JI, so we fall back to a more general acronym to distinguish it: AFS, for arithmetic frequency sequence. For example, if we wanted to move by steps of φ, like this: <math>1, 1+φ, 1+2φ, 1+3φ...</math> etc. we could have the AFSφ. Because different acronyms are used to distinguish rational (JI) tunings from general tunings which include irrational (non-JI) tunings, while the acronyms used for general tunings technically include the JI tunings, these general acronyms are more useful when reserved for non-JI tunings, and this is what is typically done. So when "irrational" is used on this page, it more accurately means "probably irrational".

== Divisions ==

So far we have looked at arithmetic tunings produced by sequencing a single step repeatedly. But if an arithmetic tuning is defined by having equal step sizes of some kind of quantity (frequency, pitch, or length), then it also follows that they can be produced by taking a larger interval and equally dividing it according to that kind of quantity.

The most common example of this type of tuning is 12edo, standard tuning, which takes the interval of the octave, and equally divides its pitch into 12 parts. For long, we could call this 12epdo, for 12 equal ''pitch'' divisions of the octave (whenever pitch is the chosen kind of quality, we can assume it, and skip pointing it out; that's why 12edo is the better name).

But it is also possible to — instead of equally dividing the octave in 12 equal parts by pitch — divide it into 12 equal parts by ''frequency'', or ''length''. In the former case, you will have 12efdo, and in the latter case, you will have 12eldo. However, that is not exactly ideal because, as with arithmetic sequences, different acronyms are used to distinguish rational (JI) tunings from irrational (non-JI) tunings, and so EFD and ELD are typically reserved for irrational tunings, such as 12efdφ. So it would be more appropriate to name these two tunings 12odo and 12udo, for otonal divisions of the octave and utonal divisions of the octave, respectively.

== Comparing arithmetic tunings ==