Editing Rhythm (section)

===Rhythm–tempo interaction===
{{Undue weight section|date=January 2021}}
One difficulty in defining rhythm is the dependence of its perception on tempo, and, conversely, the dependence of tempo perception on rhythm. Furthermore, the rhythm–tempo interaction is context dependent, as explained by Andranik Tangian using an example of the leading rhythm of "Promenade" from [[Modest Mussorgsky|Moussorgsky]]'s ''[[Pictures at an Exhibition]]'':({{sfn|Tanguiane|1993}}{{sfn|Tanguiane|1994|pages=465–502}}
{{box|type=transparent|spacing=1em|border size=0em|{{music|quarter}} {{music|quarter}} {{music|quarter}}}}{{box|type=transparent|spacing=0.2em|border size=0em|{{music|quaver}} {{music|quaver}} {{music|quaver}}}}

This rhythm is perceived as it is rather than as the first three events repeated at a double tempo (denoted as '''R012''' = repeat from 0, one time, twice faster):
{{box|type=transparent|spacing=1em|border size=0em|{{music|quarter}} {{music|quarter}} {{music|quarter}}}}{{box|type=transparent|spacing=0em|border size=0em|'''R012'''}}
However, the motive with this rhythm in the Moussorgsky's piece
{{box|type=transparent|spacing=1em|border size=0em| <sup>{{music|quarter}} <sup>{{music|quarter}}</sup></sup> <sub>{{music|quarter}}</sub>}}{{box|type=transparent|spacing=0.2em|border size=0em|<sup>{{music|quaver}} <sup>{{music|quaver}}</sup></sup> <sub>{{music|quaver}}</sub> }}
is rather perceived as a repeat
{{box|type=transparent|spacing=1em|border size=0em| <sup>{{music|quarter}} <sup>{{music|quarter}}</sup></sup> <sub>{{music|quarter}}</sub>}}{{box|type=transparent|spacing=0em|border size=0em|'''R012'''}}

This context-dependent perception of rhythm is explained by the principle of correlative perception, according to which data are perceived in the simplest way. From the viewpoint of [[Andrey Kolmogorov|Kolmogorov]]'s complexity theory, this means such a representation of the data that minimizes the amount of memory.

The example considered suggests two alternative representations of the same rhythm: as it is, and as the rhythm-tempo interaction – a two-level representation in terms of a generative rhythmic pattern and a "tempo curve". Table 1 displays these possibilities both with and without pitch, assuming that one duration requires one byte of information, one byte is needed for the pitch of one tone, and invoking the repeat algorithm with its parameters '''R012''' takes four bytes. As shown in the bottom row of the table, the rhythm without pitch requires fewer bytes if it is "perceived" as it is, without repetitions and tempo leaps. On the contrary, its melodic version requires fewer bytes if the rhythm is "perceived" as being repeated at a double tempo.

{| class="wikitable" width="auto" style="text-align:left;"
|+ Complexity of representation of time events
|-
|!style="text-align:left;" |
!colspan="2"| Rhythm only
||
!colspan="2"| Rhythm with pitch
|-
|
!Complete coding
!Coding as repeat
||
!Complete coding
!Coding as repeat
|-
|
!{{box|type=transparent|spacing=1em|border size=0em|{{music|quarter}} {{music|quarter}} {{music|quarter}}}}{{box|type=transparent|spacing=0.2em|border size=0em|{{music|quaver}} {{music|quaver}} {{music|quaver}}}}
!{{box|type=transparent|spacing=1em|border size=0em|{{music|quarter}} {{music|quarter}} {{music|quarter}}}} R012
||
!{{box|type=transparent|spacing=1em|border size=0em|<sup>{{music|quarter}} <sup>{{music|quarter}}</sup></sup> <sub>{{music|quarter}}</sub>}} {{box|type=transparent|spacing=0.2em|border size=0em|<sup>{{music|quaver}} <sup>{{music|quaver}}</sup></sup> <sub>{{music|quaver}}</sub> }}
!{{box|type=transparent|spacing=1em|border size=0em|<sup>{{music|quarter}} <sup>{{music|quarter}}</sup></sup> <sub>{{music|quarter}}</sub>}} R012
|-
! style="text-align:left;" |Complexity of rhythmic pattern
||6 bytes||3 bytes|| ||12 bytes ||6 bytes
|-
! style="text-align:left;" |Complexity of its transformation
||0 bytes||4 bytes|| ||0 bytes || 4 bytes
|-
! style="text-align:left;" |Total complexity
|| 6 bytes||7 bytes|| || 12 bytes ||10 bytes
|}
Thus, the loop of interdependence of rhythm and tempo is overcome due to the simplicity criterion, which "optimally" distributes the complexity of perception between rhythm and tempo. In the above example, the repetition is recognized because of additional repetition of the melodic contour, which results in a certain redundancy of the musical structure, making the recognition of the rhythmic pattern "robust" under tempo deviations. Generally speaking, the more redundant the "musical support" of a rhythmic pattern, the better its recognizability under augmentations and diminutions, that is, its distortions are perceived as tempo variations rather than rhythmic changes:

{{blockquote|By taking into account melodic context, homogeneity of accompaniment, harmonic pulsation, and other cues, the range of admissible tempo deviations can be extended further, yet still not preventing musically normal perception. For example, [[Alexander Scriabin|Skrjabin]]'s own performance of his ''Poem'' op. 32 no. 1 transcribed from a piano-roll recording contains tempo deviations within {{music|dottedquarter}} {{=}} 19/119, a span of 5.5 times.{{sfn|Skrjabin|1960}} Such tempo deviations are strictly prohibited, for example, in Bulgarian or Turkish music based on so-called additive rhythms with complex duration ratios, which can also be explained by the principle of correlativity of perception. If a rhythm is not structurally redundant, then even minor tempo deviations are not perceived as ''accelerando'' or ''ritardando'' but rather given an impression of a change in rhythm, which implies an inadequate perception of musical meaning.{{sfn|Tanguiane|1994|p=480}}}}