Expanding tonal space: Difference between revisions

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 ::::<math>
 r_{12edo}=integer\left (\frac{386c+50c}{100.0}  \right )\cdot 100=400</math> ¢
+== Number of distinct intervals ==
+The first five octaves of Tonal Space contain a fairly large number of intervals footed on a common tonic of 0 ¢. The intervals are well structured in rows, with each row corresponding to a mode of the overtone scale. It may be of interest to the reader to know how many ''different'' intervals are present, since some obviously occur more than once. <br>
+To find out, we will scan the Horizon Chart line by line (mode by mode), from the bottom up. Mode 1 has no intervals between the fundamental and the next octave. In Mode 2 we find a pure fifth, the third harmonic. This is the first time the pure fifth appears, and - like any other interval - it is only considered once as we scan.
+Table 1 summarizes the scanning results from Mode 1 through Mode 16.
+: <u>Table 1</u>: Count of distinct intervals depending on the highest implemented mode
+: {| class="wikitable" style="text-align:center;"
+|- style="background-color:#6b82c6; color:#484848;"
+! <span style="font-weight:normal"> Mode 1 <br /> up to <br /> Mode 16 </span> <br /> Mode
+! <span style="font-weight:normal"> Number of <br />new intervals <br />found at <br />this mode </span>
+! Aggregated <br />number <br />of different   <br />rational intervals
+! <span style="font-weight:normal"> Total number<br />of intervals<br />scanned </span>
+! <span style="font-weight:normal"> Mode<br />found in<br />octave<br />number </span>
+|-
+| style="background-color:#e2e2e2; color:#484848;" | '''16'''
+| 8
+| style="background-color:#e2e2e2;" | 79
+| 120
+| 5
+|-
+| style="background-color:#e2e2e2;" | '''15'''
+| 8
+| style="background-color:#e2e2e2;" | 71
+| 105
+| 4
+|-
+| style="background-color:#e2e2e2;" | '''14'''
+| 6
+| style="background-color:#e2e2e2;" | 63
+| 91
+| 4
+|-
+| style="background-color:#e2e2e2;" | '''13'''
+| 12
+| style="background-color:#e2e2e2;" | 57
+| 78
+| 4
+|-
+| style="background-color:#e2e2e2;" | '''12'''
+| 4
+| style="background-color:#e2e2e2;" | 45
+| 66
+| 4
+|-
+| style="background-color:#e2e2e2;" | '''11'''
+| 10
+| style="background-color:#e2e2e2;" | 41
+| 55
+| 4
+|-
+| style="background-color:#e2e2e2;" | '''10'''
+| 4
+| style="background-color:#e2e2e2;" | 31
+| 45
+| 4
+|-
+| style="background-color:#e2e2e2;" | '''9'''
+| 6
+| style="background-color:#e2e2e2;" | 27
+| 36
+| 4
+|-
+| style="background-color:#e2e2e2;" | '''8'''
+| 4
+| style="background-color:#e2e2e2;" | 21
+| 28
+| 4
+|-
+| style="background-color:#e2e2e2;" | '''7'''
+| 6
+| style="background-color:#e2e2e2;" | 17
+| 21
+| 3
+|-
+| style="background-color:#e2e2e2;" | '''6'''
+| 2
+| style="background-color:#e2e2e2;" | 11
+| 15
+| 3
+|-
+| style="background-color:#e2e2e2;" | '''5'''
+| 4 <small>''(...)''</small>
+| style="background-color:#e2e2e2;" | 9
+| 10
+| 3
+|-
+| style="background-color:#e2e2e2;" | '''4'''
+| 2  <small>''(3rd, b7th)''</small>
+| style="background-color:#e2e2e2;" | 5
+| 6
+| 3
+|-
+| style="background-color:#e2e2e2;" | '''3'''
+| 2   <small>''(4th, 6th)''</small>
+| style="background-color:#e2e2e2;" | 3
+| 3
+| 2
+|-
+| style="background-color:#e2e2e2;" | '''2'''
+| 1 <small>''(5th)''</small>
+| style="background-color:#e2e2e2;" | 1
+| 1
+| 2
+|-
+| style="background-color:#e2e2e2;" | '''1'''
+| 0
+| style="background-color:#e2e2e2;" | 0
+| 0
+| 1
+|}
 ==A variety of projections of the model==
@@ Line 99: / Line 211: @@
 Nevertheless, this representation is the basis for a variety of ''realtime'' ''operations on chords'' that a future musical instrument can apply. In addition, the Cartesian projection of tonal space (with a straight horizontal line for each mode of an overtone scale) can be easily handled in a programming environment.
+(See [[Expanding tonal space/projections|Part III]] for more on Cartesian and polar projections of tonal space.)
 ==Polar projection of tonal space==
@@ Line 115: / Line 228: @@
 [[File:Fig-5 tonal space 753i aug.png|480px|center]]
 <center><small><u>Fig.5</u>: Selection of different augmented chords </small></center>
 The model of tonal space is well suited for the design and comparison of chords. No matter what intervals we '''mark on any horizontal line''', the result will always be a chord made up of just intervals that share a common denominator. Therefore, any such chord is a local subset of the harmonic series at its proper position.
-A final example: If we want to create a major ''b''7 chord, we will find four suitable pitches in the horizontal Mode 4-line (Fig.3) from m=0 to m=3. If we want to replace the upper 7/4 interval with, say, a 9/5 interval, find the ''Least Common Denominator'' (''LCM'', which is 4*5=20 in this case), and we get a 20:25:30:36 chord, which lives in Mode 20 (not shown)  and sounds noticeably more dissonant.
+A final example: If we want to create a major ''b''7 chord, we will find four suitable pitches in the horizontal Mode 4-line (Fig.3) from m=0 to m=3. To replace the upper 7/4 interval with, say, a 9/5 interval, we determine the ''Least Common Denominator'' (''LCM'', which is 4*5=20 in this case) to get a 20:25:30:36 chord, which lives in Mode 20 (not shown) and sounds noticeably more dissonant.
 == Find out more about tonal space… ==
-====[[Expanding tonal space/planar extensions|Part II: Planar extensions]]====
+====[[Expanding tonal space/planar extensions|Part II: <span style="font-weight:normal">Planar extensions</span>]]====
+====[[Expanding tonal space/projections|Part III: <span style="font-weight:normal">Projections</span>]]====
+====[[Expanding tonal space/third dimension|Part IV: <span style="font-weight:normal">Third dimension</span>]]====
 ==See also…==