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


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==== Extending tonal space to the right ====
==== 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. This even leaves room for the more dense intervals of the 6th octave in the upper right corner.
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]]
[[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>
::<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]]'' <br>(for example starting at harmonic  ''h6'': <math>(
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> ).
\frac{6}{6}, \frac{6}{5}, \frac{6}{4}, \frac{6}{3}, \frac{6}{2})</math> ).


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