Delta scale

Image:Just diatonic semitone on C.png {{audio|Just diatonic semitone on C.mid|Play}}]]

{{multiple image

| align =

| direction = vertical

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| header = Chromatic circle

| image1 = Delta scale chromatic circle.png

| caption1 = The delta scale's approximations compared with the just values

| image2 = 12-tet scale chromatic circle for comparison with alpha scale et al.png

| caption2 = Twelve-tone equal temperament vs. just

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The δ (delta) scale is a non-octave repeating musical scale. It may be regarded as the beta scale's reciprocal, since it is "as far 'down' the (0 3 6 9) circle from α as β is 'up'".Taruskin, Richard (1996). Stravinsky and the Russian Traditions: A Biography of the Works through Mavra, p. 1394. {{ISBN|978-0-520-07099-8}}. As such it would split the minor second (presumably 16:15) into eight equal parts of approximately 14 cents each {{audio|Delta scale step on C.mid|Play}}. This would total approximately 85.7 steps per octave.

The scale step may also precisely be derived from using 50:28 (25:14, 1003.8 cents, A{{music|L}}{{music|#}}, {{audio|Septimal augmented sixth on C.mid|Play}}) to approximate the interval {{Frac|3:2|5:4}}, which equals 6:5 (E{{music|b}}, 315.64 cents, {{audio|Just minor third on C.mid|Play}}). Thus the step is approximately 13.946 cents, and there are 86.049 steps per octave.

:$\backslash begin\{align\}$

\frac{50\log_2{\left(\frac32\right)} + 28\log_2{\left(\frac54\right)} + 23\log_2{\left(\frac65\right)}}{50^2+28^2+23^2} = 0.011\,621\,2701 \\

0.011\,621\,2701 \times 1200 = 13.945\,524\,1627

\end{align}

({{audio|Delta scale step on C.mid|Play}})

The Bohlen–Pierce delta scale is based on the tritave and the 7:5:3 "wide" triad ({{audio|BP chord 357 just.mid|Play}}) and the 9:7:5 "narrow" triad ({{audio|BP 579 or 1st inversion chord.mid|Play}}) (rather than the conventional 4:5:6 triad). Notes include:[http://www.huygens-fokker.org/bpsite/tonality.html "What about BP tonality?"], The Bohlen-Pierce Site.

:1:1 {{audio|Unison on C.mid|Play}}

:25:21 {{audio|BP major second on C.mid|Play}}

:9:7 {{audio|Septimal major third on C.mid|Play}}

:75:49 {{audio|BP fifth on C.mid|Play}}

:5:3 {{audio|Just major sixth on C.mid|Play}}

:9:5 {{audio|Greater just minor seventh on C.mid|Play}}

:15:7 {{audio|Septimal diatonic semitone on C.mid|Play}}

:7:3 {{audio|Septimal minor third on C.mid|Play}}

:25:9 {{audio|Classic augmented eleventh on C.mid|Play}}

:3:1 {{audio|Tritave on C.mid|Play}}

class="wikitable" |align=center bgcolor="#ffffb4"|interval name |align=center bgcolor="#ffffb4"|size (steps) |align=center bgcolor="#ffffb4"|size (cents) |align=center bgcolor="#ffffb4"|just ratio |align=center bgcolor="#ffffb4"|just (cents) |align=center bgcolor="#ffffb4"|error

align=center|minor third |align=center|23 |align=center|321.23 |align=center|6:5 |align=center|315.64 |align=center|+5.59

align=center|major third |align=center|28 |align=center|391.06 |align=center|5:4 |align=center|386.31 |align=center|+4.75

align=center|perfect fifth |align=center|50 |align=center|698.32 |align=center|3:2 |align=center|701.96 |align=center|−3.63