User:Holger Stoltenberg/sandbox: Difference between revisions

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<center>Fig. 3: The ''Horizon Chart'': Relations of overtone scales (up to Mode 16) on a plane of tonal space</center>

Each pitch is labeled with the size of an interval in cents, measured from the tonic (0 ¢) to the corresponding pitch marker ('''+'''). Each pitch marker is connected to the ~~next~~ vertical 12edo-line by a ''delta'' ''indicator''. We define the direction and length of this indicator as the ''signed intonation'' interval of the respective pitch.

Each pitch is labeled with the size of an interval in cents, measured from the tonic (0 ¢) to the corresponding pitch marker ('''+'''). Each pitch marker is connected to the nearest vertical 12edo-line by a ''delta'' ''indicator''. We define the direction and length of this indicator as the ''signed intonation'' interval of the respective pitch.

The [[AFDO#Formula|AFDO]]-page can help to reproduce this plot:

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::*''a deviating pitch referenced by the same key descriptor''

Typically, intonation is a small interval between -50 ¢ and +50 ¢ although larger values are allowed. In our model the common tonic of all modes of the overtone scale has an intonation of 0 ¢ by definition.

Typically, intonation is a small interval between -50 ¢ and +50 ¢ although larger values are allowed. In our model, the common tonic of all modes of the overtone scale has an intonation of 0 ¢ by definition.

According to this definition'','' the upper pitch of the just major third (Mode 4) ~~above the tonic~~ has ~~a signed~~ intonation interval (distance to the ~~next~~ vertical 12edo line) of -14 ¢ .

According to this definition'','' the upper pitch of the just major third above the tonic (Mode 4, see Example 1 in the section above) has an intonation interval, which represents the ''distance to the nearest vertical 12edo line'', of -14 ¢ .

To calculate the intonation

*compute the ''remainder'' of the interval’s value in cents by a modulo division (386 ¢ ''mod''100) , the intermediate result is 86 ¢

*compute the ''remainder'' of the interval’s ''value in cents'' by a modulo division (386 ¢ ''mod''100), the intermediate result is 86 ¢

*Test: If the intermediate result is greater than 50 ¢ then subtract 100 ¢

*The test is true and the final result is -14 ¢

*The test is true and the final result is 86 ¢ -100 ¢ = -14 ¢

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

To determine the 12edo interval the intonation is applied to, 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>

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

...and according to Example 1:

::::<math>

r_{12edo}=integer\left (\frac{386c+50c}{100.0} \right )\cdot 100=400</math> ¢

==A variety of projections of the model ==

==A variety of projections of the model==

Keep in mind that the ''Horizon Chart'' (Fig.3) is just one ''graphical representation of relationships'' between pitches, musical intervals and overtone scales. More specifically, Fig.3 shows one of many useful Cartesian projections of an abstract model onto a 2D-plane.

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The center is the location of the fundamental, where Mode n=1 and m=0. This corresponds to the origin of the former Cartesian coordinate system. The mode axis runs from the center up to the north. A clockwise angle of 2π represents one octave up. Each dot represents a pitch.

==General Applicability ==

==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]]

@@ Line 41: / Line 41: @@
 <center><small><u>Fig. 3</u>: The ''Horizon Chart'': Relations of overtone scales (up to Mode 16) on a plane of tonal space</small></center>
-Each pitch is labeled with the size of an interval in cents, measured from the tonic (0 ¢) to the corresponding pitch marker (<small>'''+'''</small>). Each pitch marker is connected to the next vertical 12edo-line by a ''delta'' ''indicator''. We define the direction and length of this indicator as the ''signed intonation'' interval of the respective pitch.
+Each pitch is labeled with the size of an interval in cents, measured from the tonic (0 ¢) to the corresponding pitch marker (<small>'''+'''</small>). Each pitch marker is connected to the nearest vertical 12edo-line by a ''delta'' ''indicator''. We define the direction and length of this indicator as the ''signed intonation'' interval of the respective pitch.
 The [[AFDO#Formula|AFDO]]-page can help to reproduce this plot:
@@ Line 71: / Line 71: @@
 ::*''a deviating pitch referenced by the same key descriptor''
-Typically, intonation is a small interval between -50 ¢ and +50 ¢ although larger values are allowed. In our model the common tonic of all modes of the overtone scale has an intonation of 0 ¢ by definition.
+Typically, intonation is a small interval between -50 ¢ and +50 ¢ although larger values are allowed. In our model, the common tonic of all modes of the overtone scale has an intonation of 0 ¢ by definition.
-According to this definition'','' the upper pitch of the just major third (Mode 4) above the tonic has a signed intonation interval (distance to the next vertical 12edo line) of -14 ¢ .
+According to this definition'','' the upper pitch of the just major third above the tonic (Mode 4, see Example 1 in the section above) has an intonation interval, which represents the ''distance to the nearest vertical 12edo line'', of -14 ¢ .
 To calculate the intonation
-*compute the ''remainder'' of the interval’s value in cents by a modulo division (386 ¢ ''mod''100) , the <br>intermediate result is 86 ¢
+*compute the ''remainder'' of the interval’s ''value in cents'' by a modulo division (386 ¢ ''mod''100), the <br>intermediate result is 86 ¢
 *Test: If the intermediate result is greater than 50 ¢ then subtract 100 ¢
-*The test is true and the final result is -14 ¢
+*The test is true and the final result is 86 ¢ -100 ¢ = -14 ¢
-To determine the 12edo interval the intonation is applied to, get the original interval ''r<sub>cents </sub>'' and do some more integer arithmetic:
+To determine the 12edo interval the intonation is applied to, 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>
+r_{12edo}=integer\left (\frac{r_{cents}+50c}{100.0}  \right )\cdot 100</math> <br>
+...and according to Example 1:
 ::::<math>
 r_{12edo}=integer\left (\frac{386c+50c}{100.0}  \right )\cdot 100=400</math> ¢
-==A variety of projections of the model ==
+==A variety of projections of the model==
 Keep in mind that the ''Horizon Chart'' (Fig.3) is just one ''graphical representation of relationships'' between pitches, musical intervals and overtone scales. More specifically, Fig.3 shows one of many useful Cartesian projections of an abstract model onto a 2D-plane.
@@ Line 98: / Line 99: @@
 The center is the location of the fundamental, where Mode n=1 and m=0. This corresponds to the origin of the former Cartesian coordinate system. The mode axis runs from the center up to the north. A clockwise angle of 2π represents one octave up. Each dot represents a pitch.
-==General Applicability ==
+==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]]