User:Holger Stoltenberg/sandbox: Difference between revisions
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With a prototypical instrument in place, the search for ''intermediary [[chord]]<nowiki/>s'' in tonal music begins. We can identify (and play) supplementary consonant chords that fit seamlessly into the gaps of familiar chord progressions. Listen to the following two audio examples to get the idea... | With a prototypical instrument in place, the search for ''intermediary [[chord]]<nowiki/>s'' in tonal music begins. We can identify (and play) supplementary consonant chords that fit seamlessly into the gaps of familiar chord progressions. Listen to the following two audio examples to get the idea... | ||
{| class="wikitable" | :{| class="wikitable" | ||
! Listen to | !Listen to | ||
! Control | !Control | ||
! Info | !Info | ||
|- | |- | ||
| ...a sequence of five<br>beatless minor chords || [[File:Audio1_tonal_space_I-min+IV-min.mp3|270px]]|| <math>I</math>min, <math>I</math>min, <math>IV</math>min, <br><math>IV</math>min, <math>I</math>min | |...a sequence of five<br>beatless minor chords||[[File:Audio1_tonal_space_I-min+IV-min.mp3|270px]]||<math>I</math>min, <math>I</math>min, <math>IV</math>min, <br><math>IV</math>min, <math>I</math>min | ||
|- | |- | ||
| ...an inserted <br>''intermediary chord'' <br>on the 2nd beat || [[File:Audio2_tonal_space_I-min+I-augmin+IV-min.mp3|270px]]|| <math>I</math>min, <math>I</math>xen-augmin, <math>IV</math>min, <br><math>IV</math>min, <math>I</math>min ... | |...an inserted <br>''intermediary chord'' <br>on the 2nd beat||[[File:Audio2_tonal_space_I-min+I-augmin+IV-min.mp3|270px]]||<math>I</math>min, <math>I</math>xen-augmin, <math>IV</math>min, <br><math>IV</math>min, <math>I</math>min ... <small>''(2 times)''</small> | ||
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The [[AFDO#Formula|AFDO]]-page can help to reproduce this plot: | The [[AFDO#Formula|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 | ||
r=\frac{n+m}{n}</math> , where | r=\frac{n+m}{n}</math> , where | ||
:*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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:*if m = n, then r = 2 (1200 ¢) | :*if m = n, then r = 2 (1200 ¢) | ||
<u> | <u>Example 1</u>: A just major third relates the first element ''(m=1)'' of a Mode 4 ''(n=4)'' overtone scale to the tonic ''(m=0)'' of Mode 4 and we get | ||
::::<math> | ::::<math> | ||
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Intonation is an adjustment of pitch applied to notes - live at performance time. | Intonation is an adjustment of pitch applied to notes - live at performance time. | ||
In the context of this model, we use 12edo pitches as the reference scale for measuring intonation. To describe intonation precisely ( | In the context of this model, we use 12edo pitches as the reference scale for measuring intonation. To describe intonation precisely (but 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 pressed 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'' | ||
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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|496px|center]] | [[File:Fig-4 tonal space POLAR 149.png|496px|center]] | ||
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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. | 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: | 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]] | [[File:Fig-5 tonal space 753i aug.png|480px|center]] | ||
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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 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. | ||
== See also… == | ==See also…== | ||
Sethares, William A. ''Tuning Timbre Spectrum Scale.'' London: Springer Verlag , 1999. | Sethares, William A. ''Tuning Timbre Spectrum Scale.'' London: Springer Verlag , 1999. | ||
[p65, ''3.7. Overtone Scales''] | [p65, ''3.7. Overtone Scales''] | ||