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Expanding tonal space: Difference between revisions

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::::<math>
::::<math>
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> ¢
== 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==
==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.
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==
==Polar projection of tonal space==
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==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. <br>
[[File:Fig-5 tonal space 753i aug.png|480px|center]]
Listen to the following audio examples...
<center><small><u>Fig.5</u>: Selection of different augmented chords </small></center>
[[File:Fig-5 tonal space 753i aug.png|thumb|420px|right|<u>Fig.5</u>: Selection of different augmented chords]]
 
:{| class="wikitable"
!Chord
!Play
|-
|mode16 aug (16:20:25)<br>2 stacked pure 3rds||[[File:TS_aug_L16_p25_C_P-I_Fig-5.mp3|80px]]
|-
|mode14 aug (14:18:22)<br>sounds equal to (7:9:11)||[[File:TS_aug_L14_C_P-I_Fig-5.mp3|80px]]
|-
|mode11 aug (11:14:17)<br>no, this is not major...||[[File:TS_aug_L11-14-17_C_P-I_Fig-5.mp3|80px]]
|-
|mode10 aug (10:13:16)<br>extra wide 3rd||[[File:TS_aug_L10_C_P-I_Fig-5.mp3|80px]]
|-
|<br>mode8 maj (8:10:12)<br>'''pure major''' for reference<br><br>||[[File:TS_maj_lin_8_C_P-I_Fig-5.mp3|80px]]
|}


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.   
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.   
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== Find out more about tonal space… ==
== Find out more about tonal space… ==
====[[Expanding tonal space/planar extensions|Part II: <span style="font-weight:normal">Planar extensions</span>]]====
====[[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…==
==See also…==
Retrieved from "https://en.xen.wiki/w/Expanding_tonal_space"