Periods and generators: Difference between revisions

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It is possible to construct scales in which the period is not the octave. Of these, the most common are scales in which the period subdivides the octave, which are often called "symmetric scales" in 12-equal music theory.

One common example of a scale with a fractional-octave period is the diminished or octatonic scale, which in 12-equal is C~~-Db-~~D#-E-F#-G-A~~-Bb-~~C. Although the 12-equal tuning for this scale is 0 - 100 - 300 - 400 - 600 - 700 - 900 - 1000 - 1200 ~~cents~~, we will consider. To construct this scale, consider a generator chain of 100 cents, which we stop at 2 notes, thus yielding the following mini-chain:

One common example of a scale with a fractional-octave period is the diminished or octatonic scale, which in 12-equal is {{dash|C, D♭, D♯, E, F♯, G, A, B♭, C|s=thin|d=med}}. Although the 12-equal tuning for this scale is (in cents) {{dash|0, 100, 300, 400, 600, 700, 900, 1000, 1200|s=thin|d=med}}, we will consider. To construct this scale, consider a generator chain of 100 cents, which we stop at 2 notes, thus yielding the following mini-chain:

0 = 100 (or C - C#)

0 = 100 (or C–C♯)

If we were to tile this scale at the octave, we would arrive at the following strange 2-note periodic scale:

(…) ~~← -~~1200 ~~← -~~1100 ← 0 → 100 → 1200 → 1300 → 2400 → 2500 → (…) (or ~~… -~~ C0 ~~- Db0 -~~ C1 ~~- Db1 -~~ C2 ~~- Db2 - …~~)

{{dash|(...), −1200, −1100, 0|s=space|d=larr}} {{dash|, 100, 1200, 1300, 2400, 2500, (...)|s=space|d=rarr}} (or {{dash|..., C0, D♭0, C1, D♭1, C2, D♭2, ...|s=thin|d=med}})

However, if we were to instead tile this scale at the ~~1/4~~-octave, which means that our period is now 300 cents, we arrive at the following scale:

However, if we were to instead tile this scale at the ¼-octave, which means that our period is now 300 cents, we arrive at the following scale:

(…) ← 0 → 100 → 300 → 400 → 600 → 700 → 900 → 1000 → 1200 → (…)

{{dash|(...), 0|s=space|d=larr}} {{dash|, 100, 300, 400, 600, 700, 900, 1000, 1200, (...)|s=space|d=rarr}}

It should be noted that in the above example, 300 cents is itself generated by 100 cents, which means that strictly speaking, it is not a second "prime" generator interval. The choice of 100 cent generator was chosen for simplicity, but this is not true of all tunings for this scale – for example if our generator were 91 cents instead of 100, then the 300 cent period would not consist of 3 stacked generators anymore.

[[Category:Generator]]

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 It is possible to construct scales in which the period is not the octave. Of these, the most common are scales in which the period subdivides the octave, which are often called "symmetric scales" in 12-equal music theory.
-One common example of a scale with a fractional-octave period is the diminished or octatonic scale, which in 12-equal is C-Db-D#-E-F#-G-A-Bb-C. Although the 12-equal tuning for this scale is 0 - 100 - 300 - 400 - 600 - 700 - 900 - 1000 - 1200 cents, we will consider. To construct this scale, consider a generator chain of 100 cents, which we stop at 2 notes, thus yielding the following mini-chain:
+One common example of a scale with a fractional-octave period is the diminished or octatonic scale, which in 12-equal is {{dash|C, D&#x266D;, D&#x266F;, E, F&#x266F;, G, A, B&#x266D;, C|s=thin|d=med}}. Although the 12-equal tuning for this scale is (in cents) {{dash|0, 100, 300, 400, 600, 700, 900, 1000, 1200|s=thin|d=med}}, we will consider. To construct this scale, consider a generator chain of 100 cents, which we stop at 2 notes, thus yielding the following mini-chain:
-= 100 (or C - C#)
+= 100 (or C&ndash;C&#x266F;)
 If we were to tile this scale at the octave, we would arrive at the following strange 2-note periodic scale:
-(…) ← -1200 ← -1100 ← 0 → 100 → 1200 → 1300 → 2400 → 2500 → (…) (or … - C0 - Db0 - C1 - Db1 - C2 - Db2 - …)
+{{dash|(...), &minus;1200, &minus;1100, 0|s=space|d=larr}} {{dash|, 100, 1200, 1300, 2400, 2500, (...)|s=space|d=rarr}} (or {{dash|..., C0, D&#x266D;0, C1, D&#x266D;1, C2, D&#x266D;2, ...|s=thin|d=med}})
-However, if we were to instead tile this scale at the 1/4-octave, which means that our period is now 300 cents, we arrive at the following scale:
+However, if we were to instead tile this scale at the &frac14;-octave, which means that our period is now 300 cents, we arrive at the following scale:
-(…) ← 0 → 100 → 300 → 400 → 600 → 700 → 900 → 1000 → 1200 → (…)
+{{dash|(...), 0|s=space|d=larr}} {{dash|, 100, 300, 400, 600, 700, 900, 1000, 1200, (...)|s=space|d=rarr}}
 It should be noted that in the above example, 300 cents is itself generated by 100 cents, which means that strictly speaking, it is not a second "prime" generator interval. The choice of 100 cent generator was chosen for simplicity, but this is not true of all tunings for this scale – for example if our generator were 91 cents instead of 100, then the 300 cent period would not consist of 3 stacked generators anymore.
 [[Category:Generator]]