Interval of equivalence: Difference between revisions

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{{~~interwiki~~

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| en = ~~Equave~~

| en = Interval of equivalence

| de = ~~Äquave~~

| de = Äquivalenzintervall

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An '''interval of equivalence''' is a musical interval at which pitches are perceived or treated as functionally equivalent, despite their different absolute frequencies.

An '''interval of equivalence''' is a musical interval at which pitches are perceived or treated as functionally equivalent, despite their different absolute frequencies. When an interval is established as the equivalence interval, all pitches separated by one or more instances of this interval are considered members of the same [[pitch class]].

When an interval is established as the equivalence interval, all pitches separated by one or more instances of this interval are considered members of the same [[pitch class]].

In a [[periodic scale]], the interval of equivalence is typically the same as the [[period]] or a multiple thereof.

In a [[periodic scale]], the interval of equivalence is typically the same as the [[period]] or a multiple thereof. Almost all [[Musical notation|notation systems]] give equivalent pitches the same name.

Almost all [[Musical notation|notation systems]] give equivalent pitches the same name.

== Octave equivalence ==

The octave ([[2/1]]) is the most common interval of equivalence.

The octave ([[2/1]]) is the most common interval of equivalence. Tones separated by an octave are perceived as essentially the same. This equivalence is so strong that in most musical notation systems, notes separated by octaves share the same name. For the same reason, most [[scale]]s repeat at the octave.

Tones separated by an octave are perceived as essentially the same.

This equivalence is so strong that in most musical notation systems, notes separated by octaves share the same name.

For the same reason, most [[scale]]s repeat at the octave.

The physical basis for octave equivalence is believed to be caused by the [[harmonic series|harmonic structure]] of periodic sounds.

The physical basis for octave equivalence is believed to be caused by the [[harmonic series|harmonic structure]] of periodic sounds. When a tone with frequency ''f'' is played, it typically produces [[overtone]]s at 2''f'', 3''f'', 4''f'', etc. The first overtone (2''f'') is an octave above the fundamental, creating a consistent pattern in many natural sounds. This harmonic relationship may contribute to the perceptual significance of the octave in human hearing.

When a tone with frequency ''f'' is played, it typically produces [[overtone]]s at 2''f'', 3''f'', 4''f'', etc.

The first overtone (2''f'') is an octave above the fundamental, creating a consistent pattern in many natural sounds.

This harmonic relationship may contribute to the perceptual significance of the octave in human hearing.

The perception of octave equivalence appears to have both biological and cultural components.

The perception of octave equivalence appears to have both biological and cultural components. Studies have shown that various non-human animals, including rhesus monkeys and rats, experience octave equivalence to a certain extent.<ref>Hoeschele M. [https://doi.org/10.3819/CCBR.2017.120002 ''Animal Pitch Perception: Melodies and Harmonies'']. Comp Cogn Behav Rev. 2017</ref> However, the degree of octave equivalence perception varies across cultures, particularly in those with non-Western musical traditions.<ref>Jacoby, Nori et al. [https://doi.org/10.1016/j.cub.2019.08.020 ''Universal and Non-universal Features of Musical Pitch Perception Revealed by Singing'']. Current Biology, 2019.</ref>

Studies have shown that various non-human animals, including rhesus monkeys and rats, experience octave equivalence to a certain extent.<ref>Hoeschele M. [https://doi.org/10.3819/CCBR.2017.120002 ''Animal Pitch Perception: Melodies and Harmonies'']. Comp Cogn Behav Rev. 2017</ref>

However, the degree of octave equivalence perception varies across cultures, particularly in those with non-Western musical traditions.<ref>Jacoby, Nori et al. [https://doi.org/10.1016/j.cub.2019.08.020 ''Universal and Non-universal Features of Musical Pitch Perception Revealed by Singing'']. Current Biology, 2019.</ref>

== Non-octave equivalence ==

When the interval of equivalence is ~~specifically not an~~ octave, it is called the '''equave''' (a portmanteau of "equivalence" and "octave")~~. (The app Scala refers to this as a~~ '''formal octave'''<ref>Op de Coul M. ''Scala help'' https://www.huygens-fokker.org/scala/help.htm</ref>).

When the interval of equivalence is allowed to be a non-octave interval, it is called the '''equave''' (a portmanteau of "equivalence" and "octave") or '''formal octave'''<ref>Op de Coul M. ''Scala help'' https://www.huygens-fokker.org/scala/help.htm</ref>. While the octave is the predominant interval of equivalence in most musical traditions, alternative equivalence intervals have been explored. A famous example is the [[tritave]] in [[Bohlen–Pierce]] and related systems.

While the octave is the predominant interval of equivalence in most musical traditions, alternative equivalence intervals have been explored.

A famous example is the [[tritave]] in [[Bohlen–Pierce]] and related systems.

=== Pseudo-octave ===

A '''pseudo-octave''' is an interval that is not exactly [[2/1]], but close enough to still be treated as an octave.

[[Stretched tuning|Stretched or compressed tunings]] may be said to have pseudo-octaves, as do many non-octave scales.

[[Stretched tuning|Stretched or compressed tunings]] may be said to have pseudo-octaves, as do many non-octave scales. For example, the [[Carlos Beta]] scale has a pseudo-octave of 1212.8{{c}} at 19 steps.

For example, the [[Carlos Beta]] scale has a pseudo-octave of 1212.8{{c}} at 19 steps.

== Examples ==

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-{{interwiki
+{{Interwiki
-| en = Equave
+| en = Interval of equivalence
-| de = Äquave
+| de = Äquivalenzintervall
 | es =
 | ja =
 }}
-An '''interval of equivalence''' is a musical interval at which pitches are perceived or treated as functionally equivalent, despite their different absolute frequencies.
+An '''interval of equivalence''' is a musical interval at which pitches are perceived or treated as functionally equivalent, despite their different absolute frequencies. When an interval is established as the equivalence interval, all pitches separated by one or more instances of this interval are considered members of the same [[pitch class]].
-When an interval is established as the equivalence interval, all pitches separated by one or more instances of this interval are considered members of the same [[pitch class]].
-In a [[periodic scale]], the interval of equivalence is typically the same as the [[period]] or a multiple thereof.
+In a [[periodic scale]], the interval of equivalence is typically the same as the [[period]] or a multiple thereof. Almost all [[Musical notation|notation systems]] give equivalent pitches the same name.
-Almost all [[Musical notation|notation systems]] give equivalent pitches the same name.
 == Octave equivalence ==
-The octave ([[2/1]]) is the most common interval of equivalence.
+The octave ([[2/1]]) is the most common interval of equivalence. Tones separated by an octave are perceived as essentially the same. This equivalence is so strong that in most musical notation systems, notes separated by octaves share the same name. For the same reason, most [[scale]]s repeat at the octave.
-Tones separated by an octave are perceived as essentially the same.
-This equivalence is so strong that in most musical notation systems, notes separated by octaves share the same name.
-For the same reason, most [[scale]]s repeat at the octave.
-The physical basis for octave equivalence is believed to be caused by the [[harmonic series|harmonic structure]] of periodic sounds.
+The physical basis for octave equivalence is believed to be caused by the [[harmonic series|harmonic structure]] of periodic sounds. When a tone with frequency ''f'' is played, it typically produces [[overtone]]s at 2''f'', 3''f'', 4''f'', etc. The first overtone (2''f'') is an octave above the fundamental, creating a consistent pattern in many natural sounds. This harmonic relationship may contribute to the perceptual significance of the octave in human hearing.
-When a tone with frequency ''f'' is played, it typically produces [[overtone]]s at 2''f'', 3''f'', 4''f'', etc.
-The first overtone (2''f'') is an octave above the fundamental, creating a consistent pattern in many natural sounds.
-This harmonic relationship may contribute to the perceptual significance of the octave in human hearing.
-The perception of octave equivalence appears to have both biological and cultural components.
+The perception of octave equivalence appears to have both biological and cultural components. Studies have shown that various non-human animals, including rhesus monkeys and rats, experience octave equivalence to a certain extent.<ref>Hoeschele M. [https://doi.org/10.3819/CCBR.2017.120002 ''Animal Pitch Perception: Melodies and Harmonies'']. Comp Cogn Behav Rev. 2017</ref> However, the degree of octave equivalence perception varies across cultures, particularly in those with non-Western musical traditions.<ref>Jacoby, Nori et al. [https://doi.org/10.1016/j.cub.2019.08.020 ''Universal and Non-universal Features of Musical Pitch Perception Revealed by Singing'']. Current Biology, 2019.</ref>
-Studies have shown that various non-human animals, including rhesus monkeys and rats, experience octave equivalence to a certain extent.<ref>Hoeschele M. [https://doi.org/10.3819/CCBR.2017.120002 ''Animal Pitch Perception: Melodies and Harmonies'']. Comp Cogn Behav Rev. 2017</ref>
-However, the degree of octave equivalence perception varies across cultures, particularly in those with non-Western musical traditions.<ref>Jacoby, Nori et al. [https://doi.org/10.1016/j.cub.2019.08.020 ''Universal and Non-universal Features of Musical Pitch Perception Revealed by Singing'']. Current Biology, 2019.</ref>
 == Non-octave equivalence ==
-When the interval of equivalence is specifically not an octave, it is called the '''equave''' (a portmanteau of "equivalence" and "octave"). (The app Scala refers to this as a '''formal octave'''<ref>Op de Coul M. ''Scala help'' https://www.huygens-fokker.org/scala/help.htm</ref>).
+When the interval of equivalence is allowed to be a non-octave interval, it is called the '''equave''' (a portmanteau of "equivalence" and "octave") or '''formal octave'''<ref>Op de Coul M. ''Scala help'' https://www.huygens-fokker.org/scala/help.htm</ref>. While the octave is the predominant interval of equivalence in most musical traditions, alternative equivalence intervals have been explored. A famous example is the [[tritave]] in [[Bohlen–Pierce]] and related systems.
-While the octave is the predominant interval of equivalence in most musical traditions, alternative equivalence intervals have been explored.
-A famous example is the [[tritave]] in [[Bohlen–Pierce]] and related systems.
 === Pseudo-octave ===
 {{Wikipedia|Pseudo-octave}}
 A '''pseudo-octave''' is an interval that is not exactly [[2/1]], but close enough to still be treated as an octave.
-[[Stretched tuning|Stretched or compressed tunings]] may be said to have pseudo-octaves, as do many non-octave scales.
+[[Stretched tuning|Stretched or compressed tunings]] may be said to have pseudo-octaves, as do many non-octave scales. For example, the [[Carlos Beta]] scale has a pseudo-octave of 1212.8{{c}} at 19 steps.
-For example, the [[Carlos Beta]] scale has a pseudo-octave of 1212.8{{c}} at 19 steps.
 == Examples ==