Expanding tonal space/planar extensions: Difference between revisions

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This is '''Part II''' of a small series of articles discussing the ''model of tonal space''.

On this subpage, we explore what happens beyond the represented boundaries of tonal space (from [[Expanding tonal space|'''Part I''']]) as we extend the plane in different directions.~~ ~~

On this subpage, we explore what happens beyond the represented boundaries of tonal space (from [[Expanding tonal space|'''Part I''']]) as we extend the plane in different directions.

~~'''** THIS IS A WORK IN PROGRESS 1.Feb 2025 **'''~~

__TOC__

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It is worth noting that the Genesis scale covers all intervals of tonal space from Mode 1 to Mode 11 without a gap (with the exception of intervals containing prime factors of 13 or greater, which was a design choice). The marker for these 11-limit intervals (Fig.2) is a small plus sign (+). The ♦-markers indicate Partch’s ''multiple-number ratios'' beyond the 11-limit, x-markers indicate unimplemented intervals.

== Horizontal extension of tonal space (interval axis)==

==== Horizontal tiling ====

Since many [[Overtone scale|overtone scales]] are defined as ''octave repeating'', identical tiles placed along the horizontal interval axis can illustrate the use of ''cross-octave intervals'' or the possibility of rendering intervals in the next higher (blue frame) or lower (green frame) octave (Fig.3).

[[File:Fig-3_Extending_0847g_tiles.png|468px|center]]

::<center>Fig.3: Three copies of a Mode 16 Horizon Chart along the interval axis</center>

==== Extending tonal space to the right ====

As we know, interval spacing actually gets narrower in the next higher octave of an overtone scale. For a more natural appearance of adjacent octaves, we can shift the two corresponding frames (e.g. orange and blue) vertically by one octave (Fig.4). This even leaves room for the more dense intervals of the 6th octave in the upper right corner.

[[File:Fig-4_Extending_450_1-32_2oct.png|640px|center]]

::<center>Fig.4: Two octaves view of tonal space</center>

The ''slanted'' fine blue lines connect intervals that share a common numerator, since Harry Partch also known as ''[[Otonality and utonality|utonalities]]'' (for example starting at harmonic ''h6'': <math>(

\frac{6}{6}, \frac{6}{5}, \frac{6}{4}, \frac{6}{3}, \frac{6}{2})</math> ).

[[File:Fig-5_Partch_Diamond_Limit_11_V_11_480dpi.png|thumb|220px|Fig.5: 11-limit tonality diamond mapped to tonal space]]

To the editor's surprise, Partch's 11-limit [[wikipedia:Tonality_diamond|tonality diamond]] can be mapped seamlessly to tonal space right at the boundary between two octaves (Fig.5).

== Extending tonal space downward ==

By extending tonal space downward, we enter unfamiliar territory.

Remember that the [[frequency ratio]] ''r'' of the m-th element of an overtone scale is...

:::<math>

r=\frac{n+m}{n}</math> , where

:*r is a rational frequency ratio

:*n is the mode of an overtone scale, typically an integer

:*m addresses (indexes, counts) the elements of each overtone scale in horizontal direction from the tonic (left, starting at 0) to the right.

So far we have normalized the tonic ''of any mode n of an overtone scale'' to 1.0 by dividing n by itself, such as

<math> r=\frac{3}{3}</math> = 1.0 for Mode 3 and <math> r=\frac{2}{2}</math> = 1.0 for Mode 2 and so on. The result of these operations is that in tonal space all modes refer to a common tonic of 0 cents.

Obviously, we run out of integers, when we try to address modes of overtone scales smaller than 1. Mode 0 is undefined, and negative mode numbers make no sense.

A possible solution to this problem is to represent the octave numbers in exponential form such as

::23 for the fourth octave1) (Mode 8)

::22 for the third octave (Mode 4)

::21 for the second octave (Mode 2)

::20 for the first octave (Mode 1)

1 ''Note that by convention, the octave number is 1 greater than the exponent''

The next logical steps for the area of tonal space below Mode 1 are

::2-1 for a ''fractional'' Mode 1/2 = 0.5 2)

:: 2-2 for a ''fractional'' Mode 1/4 = 0.25

2 ''The exponent <math>x</math> for an arbitrary given fractional mode can be calculated as ''<math>x=\frac{1}{ln(2)}\cdot ln(n)</math>

So we extend the plane of tonal space downward by ''fractional modes'' < 1 and

enter the field of tritaves, pentaves and doubled octaves (also known as tetraves).

[[File:Fig-6_Extending_LE-1_4562.png|thumb|520px|center|Fig.6: Extending tonal space downward]]

==== Tritave ====

In a Mode '''2-1''' (=0.5) overtone scale the first interval above the fundamental is the [[Tritave|tritave]] (3/1, Fig.6), which substitutes the [[Octave|octave]] (2/1), the first interval in the Mode 1 overtone scale. Incrementing ''m'' in steps of 1, we find only odd harmonics (e.g. 3, 5, 7, 9, 11) and no octaves in this fractional mode of the overtone scale.

::::<math>

r=\frac{0.5+1}{0.5}=\frac{3}{1}</math>

::::<math>

r_{cents}= ln(r)\cdot\frac{1200}{ln(2)} \approx 1902</math> ¢

==== Pentave ====

In a Mode '''2-2''' (=0.25) overtone scale the first interval above the fundamental is the [[pentave]] (5/1), which substitutes the octave (2/1), the first interval in the Mode 1 overtone scale. This fractional mode of the overtone scale contains no octaves. It contains only odd harmonics with a spacing of 4 (e.g. 5, 9, 13, 17, 21, 25).

::::<math>

r=\frac{0.25+1}{0.25}=\frac{5}{1}</math>

::::<math>

r_{cents}= ln(r)\cdot\frac{1200}{ln(2)} \approx 2786</math> ¢

==== Double octave ====

In a Mode'''''(1/3)''''' overtone scale the first interval above the fundamental is the [[4/1|double octave]] (4/1), sometimes called the ''tetrave''. It substitutes the octave (2/1), the first interval in the Mode 1 overtone scale.

This fractional mode of the overtone scale contains every second octave. Octaves with 1200 ¢, 3600 ¢,… are not present in this mode. It contains harmonics with a spacing of 3 (e.g. 4, 7, 10, 13, 16).

::::<math>

r=\frac{\frac{1}{3}+1}{\frac{1}{3}}=\frac{4}{1}</math>

::::<math>

r_{cents}= ln(r)\cdot\frac{1200}{ln(2)} = 2400</math> ¢

To find the exponent x for the mode number in exponential form we solve <math>2^x\,=\frac{1}{3}</math> for x:

::::<math>x = \frac{ln(\frac{1}{3})}{ln(2)} \approx -1.585</math>

== Find out more about tonal space… ==

==== [[Expanding tonal space|Part I: Expanding tonal space]] ====

==== [[Expanding tonal space/projections|Part III: Projections]] ====

==== [[Expanding tonal space/third dimension|Part IV: Third dimension]] ====

== See also... ==

[[OS|Otonal sequence]] (OS)

[[Category:Pitch space]]

[[Category:Just intonation]]

[[Category:Scale]]

[[Category:Harmonic series]]

@@ Line 1: / Line 1: @@
+{{breadcrumb}}
 This is '''Part II''' of a small series of articles discussing the ''model of tonal space''.<br>
-On this subpage, we explore what happens beyond the represented boundaries of tonal space (from [[Expanding tonal space|'''Part I''']]) as we extend the plane in different directions.<br>
+On this subpage, we explore what happens beyond the represented boundaries of tonal space (from [[Expanding tonal space|'''Part I''']]) as we extend the plane in different directions.
-'''** THIS IS A WORK IN PROGRESS 1.Feb 2025 **'''
 __TOC__
@@ Line 19: / Line 19: @@
 It is worth noting that the Genesis scale covers all intervals of tonal space from Mode 1 to Mode 11 without a gap (with the exception of intervals containing prime factors of 13 or greater, which was a design choice). The marker for these 11-limit intervals (Fig.2) is a small plus sign (+). The ♦-markers indicate Partch’s ''multiple-number ratios'' beyond the 11-limit, x-markers indicate unimplemented intervals.
+== Horizontal extension of tonal space (interval axis)==
+==== Horizontal tiling ====
+Since many [[Overtone scale|overtone scales]] are defined as ''octave repeating'', identical tiles placed along the horizontal interval axis can illustrate the use of ''cross-octave intervals'' or the possibility of rendering intervals in the next higher (blue frame) or lower (green frame) octave (Fig.3).
+[[File:Fig-3_Extending_0847g_tiles.png|468px|center]]
+::<center><small><u>Fig.3</u>: Three copies of a Mode 16 Horizon Chart along the interval axis</small></center>
+==== Extending tonal space to the right ====
+As we know, interval spacing actually gets narrower in the next higher octave of an overtone scale. For a more natural appearance of adjacent octaves, we can shift the two corresponding frames (e.g. orange and blue) vertically by one octave (Fig.4). This even leaves room for the more dense intervals of the 6th octave in the upper right corner.
+[[File:Fig-4_Extending_450_1-32_2oct.png|640px|center]]
+::<center><small><u>Fig.4</u>: Two octaves view of tonal space</small></center>
+The ''slanted'' fine blue lines connect intervals that share a common numerator, since Harry Partch also known as ''[[Otonality and utonality|utonalities]]'' (for example starting at harmonic  ''h6'': <math>(
+\frac{6}{6}, \frac{6}{5}, \frac{6}{4}, \frac{6}{3}, \frac{6}{2})</math> ).
+[[File:Fig-5_Partch_Diamond_Limit_11_V_11_480dpi.png|thumb|220px|<u>Fig.5</u>: 11-limit tonality diamond mapped to tonal space]]
+To the editor's surprise, Partch's 11-limit [[wikipedia:Tonality_diamond|tonality diamond]] can be mapped seamlessly to tonal space right at the boundary between two octaves (Fig.5).
+== Extending tonal space downward ==
+By extending tonal space downward, we enter unfamiliar territory.
+Remember that the [[frequency ratio]] ''r'' of the m-th element of an overtone scale is...
+:::<math>
+r=\frac{n+m}{n}</math>  , where
+:*r is a rational frequency ratio
+:*n is the mode of an overtone scale, typically an integer
+:*m addresses (indexes, counts) the elements of each overtone scale in horizontal direction from the tonic (left, starting at 0) to the right.
+So far we have normalized the tonic ''of any mode n of an overtone scale'' to 1.0 by dividing n by itself, such as <br>
+<math> r=\frac{3}{3}</math> = 1.0 for Mode 3 and <math> r=\frac{2}{2}</math> = 1.0 for Mode 2 and so on. The result of these operations is that in tonal space all modes refer to a common tonic of 0 cents.
+Obviously, we run out of integers, when we try to address modes of overtone scales smaller than 1. Mode 0 is undefined, and negative mode numbers make no sense.
+A possible solution to this problem is to represent the octave numbers in exponential form such as
+::2<sup>3</sup> for the fourth octave<sup>1)</sup> (Mode 8)<br>
+::2<sup>2</sup> for the third octave (Mode 4)<br>
+::2<sup>1</sup> for the second octave (Mode 2)<br>
+::2<sup>0</sup> for the first octave (Mode 1)<br>
+<sup>1</sup> ''Note that by convention, the octave number is 1 greater than the exponent''
+The next logical steps for the area of tonal space below Mode 1 are
+::2<sup>-1</sup> for a ''fractional'' Mode 1/2 = 0.5 <sup> 2)</sup>
+:: 2<sup>-2</sup> for a ''fractional'' Mode 1/4 = 0.25
+<sup>2</sup> ''The exponent <math>x</math> for an arbitrary given fractional mode can be calculated as ''<math>x=\frac{1}{ln(2)}\cdot ln(n)</math><br>
+So we extend the plane of tonal space downward by ''fractional modes'' < 1 and
+enter the field of tritaves, pentaves and doubled octaves (also known as tetraves).
+[[File:Fig-6_Extending_LE-1_4562.png|thumb|520px|center|<u>Fig.6</u>: Extending tonal space downward]]
+==== Tritave ====
+In a Mode '''2<sup>-1'''</sup> (=0.5) overtone scale the first interval above the fundamental is the [[Tritave|tritave]] (3/1, Fig.6), which substitutes the [[Octave|octave]] (2/1), the first interval in the Mode 1 overtone scale. Incrementing ''m'' in steps of 1, we find only odd harmonics (e.g. 3, 5, 7, 9, 11) and no octaves in this fractional mode of the overtone scale.
+::::<math>
+r=\frac{0.5+1}{0.5}=\frac{3}{1}</math>
+::::<math>
+r_{cents}= ln(r)\cdot\frac{1200}{ln(2)} \approx 1902</math> ¢
+==== Pentave ====
+In a Mode '''2<sup>-2'''</sup> (=0.25) overtone scale the first interval above the fundamental is the [[pentave]] (5/1), which substitutes the octave (2/1), the first interval in the Mode 1 overtone scale. This fractional mode of the overtone scale contains no octaves. It contains only odd harmonics with a spacing of 4 (e.g. 5, 9, 13, 17, 21, 25).
+::::<math>
+r=\frac{0.25+1}{0.25}=\frac{5}{1}</math>
+::::<math>
+r_{cents}= ln(r)\cdot\frac{1200}{ln(2)} \approx 2786</math> ¢
+==== Double octave ====
+In a Mode'''''(1/3)''''' overtone scale the first interval above the fundamental is the [[4/1|double octave]] (4/1), sometimes called the ''tetrave''. It substitutes the octave (2/1), the first interval in the Mode 1 overtone scale.
+This fractional mode of the overtone scale contains every second octave. Octaves with 1200 ¢, 3600 ¢,… are not present in this mode. It contains harmonics with a spacing of 3 (e.g. 4, 7, 10, 13, 16).
+::::<math>
+r=\frac{\frac{1}{3}+1}{\frac{1}{3}}=\frac{4}{1}</math>
+::::<math>
+r_{cents}= ln(r)\cdot\frac{1200}{ln(2)} = 2400</math> ¢
+To find the exponent x for the mode number in exponential form we solve <math>2^x\,=\frac{1}{3}</math> for x:
+::::<math>x = \frac{ln(\frac{1}{3})}{ln(2)} \approx -1.585</math>
+== Find out more about tonal space… ==
+==== [[Expanding tonal space|Part I: <span style="font-weight:normal">Expanding tonal space</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... ==
+[[OS|Otonal sequence]] (OS)
+[[Category:Pitch space]]
+[[Category:Just intonation]]
+[[Category:Scale]]
+[[Category:Harmonic series]]