Expanding tonal space: Difference between revisions

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==The mode axis==

The vertical axis of our model of tonal space indicates ''modes'' of the [[harmonic series]]. All modes start at the same normalized frequency (or pitch) of 0 cents. This means that the ''tonics'' of all of these overtone scales share exactly the same pitch (Fig. 2), which can be chosen arbitrarily.

The vertical axis of our model of tonal space indicates ''[[Mode|modes]]'' of the [[harmonic series]]. All modes start at the same normalized frequency (or pitch) of 0 cents. This means that the ''tonics'' of all of these overtone scales share exactly the same pitch (Fig. 2), which can be chosen arbitrarily.

Since each mode (row) begins with a member of the harmonic series, we can space adjacent modes by corresponding rational intervals for plotting on the logarithmic vertical axis.

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==== Determine the nearest 12edo interval ====

To determine the ~~nearest~~ 12edo interval ~~the~~ intonation is applied to, get the original interval ''rcents '' and do some integer arithmetic:

In order to determine the closest 12edo interval (''r12edo '') to which intonation is to be applied, get the original interval ''rcents '' and do some integer arithmetic:

::::<math>

r_{12edo}=integer\left (\frac{r_{cents}+50c}{100.0} \right )\cdot 100</math>

... according to [[#example01|Example 1]]:

::::<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.

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.

: Table 1: Count of distinct intervals depending on the highest implemented mode

: {| class="wikitable" style="text-align:center;"

|- style="background-color:#6b82c6; color:#484848;"

! Mode 1 up to Mode 16 Mode

! Number of new intervals found at this mode

! Aggregated number of different rational intervals

! Total number of intervals scanned

! Mode found in octave number

|-

| 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 ''(...)''

| style="background-color:#e2e2e2;" | 9

| 10

| 3

|-

| style="background-color:#e2e2e2;" | '''4'''

| 2 ''(3rd, b7th)''

| style="background-color:#e2e2e2;" | 5

| 6

| 3

|-

| style="background-color:#e2e2e2;" | '''3'''

| 2 ''(4th, 6th)''

| style="background-color:#e2e2e2;" | 3

| 3

| 2

|-

| style="background-color:#e2e2e2;" | '''2'''

| 1 ''(5th)''

| 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==

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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==

In polar projection ~~of the model,~~ the horizontal ''mode'' lines form circles:

In polar projection the horizontal ''mode'' lines form circles:

[[File:Fig-4 tonal space POLAR 149.png|496px|center]]

<center>Fig.4: A plane of tonal space in polar projection (up to Mode 16)</center>

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The labeled spokes reveal a hidden symmetry of tonal space that is not apparent in Cartesian projection.

Note that a line drawn from the center to the south (at an angle of π) represents an irrational interval with a frequency ratio of √2 ≈ 1.414..., a tritone of 600 ¢. Unsurprisingly, no rational interval fits exactly on this line.

==General applicability==

In the model discussed so far a ''chord'' is composed of at least two stacked intervals with frequency ratios taken from the harmonic series in ascending order. The chord should be footed on the tonic of the particular mode. Skipped harmonics within a chord may remain mute. Fig.5 shows a comparison of four augmented chords that sound quite different:

[[File:Fig-5 tonal space 753i aug.png|480px|center]]

<center>Fig.5: Selection of different augmented chords </center>

The model of tonal space is well suited for the design and comparison of chords. No matter what intervals you '''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 ~~you~~ want to create a major ''b''7 chord, ~~you~~ will find four suitable pitches in the horizontal Mode 4-line (Fig.3) from m=0 to m=3. ~~If you 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 you~~ get a 20:25:30:36 chord, which lives in Mode 20 (not shown) ~~and~~ sounds noticeably more dissonant.

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. 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: Planar extensions]]====

====[[Expanding tonal space/projections|Part III: Projections]]====

====[[Expanding tonal space/third dimension|Part IV: Third dimension]]====

==See also…==

@@ Line 29: / Line 29: @@
 ==The mode axis==
-The vertical axis of our model of tonal space indicates ''modes'' of the [[harmonic series]]. All modes start at the same normalized frequency (or pitch) of 0 cents. This means that the ''tonics'' of all of these overtone scales share exactly the same pitch (Fig. 2), which can be chosen arbitrarily.
+The vertical axis of our model of tonal space indicates ''[[Mode|modes]]'' of the [[harmonic series]]. All modes start at the same normalized frequency (or pitch) of 0 cents. This means that the ''tonics'' of all of these overtone scales share exactly the same pitch (Fig. 2), which can be chosen arbitrarily.
 Since each mode (row) begins with a member of the harmonic series, we can space adjacent modes by corresponding rational intervals for plotting on the logarithmic vertical axis.
@@ Line 87: / Line 87: @@
 ==== Determine the nearest 12edo interval ====
-To determine the nearest 12edo interval the intonation is applied to, get the original interval ''r<sub>cents </sub>'' and do some integer arithmetic:
+In order to determine the closest 12edo interval (''r<sub>12edo </sub>'') to which intonation is to be applied, get the original interval ''r<sub>cents </sub>'' and do some integer arithmetic:
 ::::<math>
 r_{12edo}=integer\left (\frac{r_{cents}+50c}{100.0}  \right )\cdot 100</math> <br>
 ... according to [[#example01|Example 1]]:
 ::::<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 98: / 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==
-In polar projection of the model, the horizontal ''mode'' lines form circles:
+In polar projection the horizontal ''mode'' lines form circles:
 [[File:Fig-4 tonal space POLAR 149.png|496px|center]]
 <center><small><u>Fig.4</u>: A plane of tonal space in polar projection (up to Mode 16)</small></center>
@@ Line 108: / Line 222: @@
 The labeled spokes reveal a hidden symmetry of tonal space that is not apparent in Cartesian projection.
 Note that a line drawn from the center to the south (at an angle of π) represents an irrational interval with a frequency ratio of √2 ≈ 1.414..., a tritone of 600 ¢. Unsurprisingly, no rational interval fits exactly on this line.
 ==General applicability==
 In the model discussed so far a ''chord'' is composed of at least two stacked intervals with frequency ratios taken from the harmonic series in ascending order. The chord should be footed on the tonic of the particular mode. Skipped harmonics within a chord may remain mute. Fig.5 shows a comparison of four augmented chords that sound quite different:
 [[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 you '''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 you want to create a major ''b''7 chord, you will find four suitable pitches in the horizontal Mode 4-line (Fig.3) from m=0 to m=3. If you 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 you get a 20:25:30:36 chord, which lives in Mode 20 (not shown)  and sounds noticeably more dissonant.
+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. 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…==