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> ¢

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==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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[[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.

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 ~~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.

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.

== Find out more about tonal space… ==

@@ 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> ¢
@@ Line 100: / Line 101: @@
 ==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 113: / Line 114: @@
 [[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.
+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 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.
+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.
 == Find out more about tonal space… ==