User:Holger Stoltenberg/sandbox: Difference between revisions
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[[File:Fig-1_tonal_space_0846c.png|480px|center]] | [[File:Fig-1_tonal_space_0846c.png|480px|center]] | ||
<center><small><u>Fig.1</u>: One octave (from the first harmonic h1 to the second harmonic h2)</small></center> | |||
==The mode axis== | ==The mode axis== | ||
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[[File:Fig-2_tonal_space_0851f.png|480px|center]] | [[File:Fig-2_tonal_space_0851f.png|480px|center]] | ||
<center><small><u>Fig. 2</u>: Overtone scales from Mode 1 (''h1'') to Mode 4 ''(h4)'' on a 2D-plane of tonal space</small></center> | |||
== The ''Horizon Chart'' == | ==The ''Horizon Chart''== | ||
The result of visualizing more modes of the overtone scale (up to Mode 16) is shown in Fig. 3: | The result of visualizing more modes of the overtone scale (up to Mode 16) is shown in Fig. 3: | ||
[[File:Fig-3_tonal_space_0844g.png|458px|center]] | [[File:Fig-3_tonal_space_0844g.png|458px|center]] | ||
<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 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. | ||
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The [[AFDO]]-page can help to reproduce this plot: | The [[AFDO]]-page can help to reproduce this plot: | ||
: ''Within each period of any n-afdo system, the [[frequency ratio]] r of the m-th degree is'' | :''Within each period of any n-afdo system, the [[frequency ratio]] r of the m-th degree is'' | ||
::::<math>\displaystyle | ::::<math>\displaystyle | ||
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:*r is a rational frequency ratio which – after conversion to cents –<br> is displayed against the horizontal interval axis of tonal space | :*r is a rational frequency ratio which – after conversion to cents –<br> is displayed against the horizontal interval axis of tonal space | ||
:*n is the mode of an overtone scale, plotted on the vertical axis | :* n is the mode of an overtone scale, plotted on the vertical axis | ||
:*m addresses (indexes, counts) the elements of each overtone scale in horizontal direction from the tonic (left, starting at 0) to the right. | :*m addresses (indexes, counts) the elements of each overtone scale in horizontal direction from the tonic (left, starting at 0) to the right. | ||
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In the context of this model, we use 12edo pitches as the reference scale for measuring intonation. To describe intonation precisely (and without reference to concert pitch or absolute frequency), we define | In the context of this model, we use 12edo pitches as the reference scale for measuring intonation. To describe intonation precisely (and without reference to concert pitch or absolute frequency), we define | ||
:: ''Intonation is the signed interval between...'' | ::''Intonation is the signed interval between...'' | ||
::* ''a pitch, generated by a key with a given key descriptor (such as A3, B3, C4, C#4, ...), <br>that rings exactly in tune with 12-tone equal temperament (12edo) and'' | ::*''a pitch, generated by a key with a given key descriptor (such as A3, B3, C4, C#4, ...), <br>that rings exactly in tune with 12-tone equal temperament (12edo) and'' | ||
::* ''a deviating pitch referenced by the same key descriptor'' | ::*''a deviating pitch referenced by the same key descriptor'' | ||
Typically, intonation is a small interval between -50 ¢ and +50 ¢ although larger values are allowed. 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. The common tonic of all modes of the overtone scale has an intonation of 0 ¢ by definition. | ||
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To calculate the intonation | To calculate the intonation | ||
* compute the ''remainder'' of the interval’s value in cents by a modulo division (386 ¢ ''mod''100) – <br>the intermediate result is 86 ¢ | *compute the ''remainder'' of the interval’s value in cents by a modulo division (386 ¢ ''mod''100) – <br>the intermediate result is 86 ¢ | ||
* Test: If the intermediate result is greater than 50 ¢ then subtract 100 ¢ | *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 -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 more integer arithmetic: | ||
::::<math> | ::::<math> | ||
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r_{12edo}=integer\left (\frac{386c+50c}{100.0} \right )\cdot 100=400</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. | 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. | ||
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. | 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. | ||
== Polar projection of tonal space == | ==Polar projection of tonal space == | ||
In polar projection of the model, the horizontal ''mode'' lines form circles: | In polar projection of the model, the horizontal ''mode'' lines form circles: | ||
[[File:Fig-4 tonal space POLAR 149.png|458px|center]] | [[File:Fig-4 tonal space POLAR 149.png|458px|center]] | ||
<center><small><u>Fig.4</u>: A plane of tonal space in polar projection (up to Mode 16)</small></center> | |||
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. | 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== | |||
In this model 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 comparing chords. No matter what intervals you '''mark on any horizontal line''', the result will always be a chord made up of rational intervals that share a common denominator. Such a chord is therefore a subset of the harmonic series. | |||
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. | |||