9/8: Difference between revisions
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{{Infobox Interval | {{Infobox Interval | ||
| Name = whole tone, major second | | Name = Pythagorean whole tone, Pythagorean major second | ||
| Color name = w2, wa 2nd | | Color name = w2, wa 2nd | ||
| Sound = jid_9_8_pluck_adu_dr220.mp3 | | Sound = jid_9_8_pluck_adu_dr220.mp3 | ||
| Line 7: | Line 7: | ||
{{Wikipedia|Major second}} | {{Wikipedia|Major second}} | ||
'''9/8''' | '''9/8''', the '''Pythagorean whole tone''' or '''major second''', is an interval measuring approximately 203.9¢. It can be arrived at by stacking two just perfect fifths ([[3/2]]) and reducing the result by one octave. However, it is also a relatively low overtone in its own right, octave-reduced. It can be treated as a dissonance or a consonance, depending on compositional context, though because of its relatively close proximity to the [[unison]], it is the largest [[superparticular]] interval known to cause crowding, which lends more to it being considered a type of dissonance- at least in historical Western Classical traditions and in the xenharmonic traditions derived from them. | ||
Two 9/8's stacked produce [[81/64]], the Pythagorean major third, a rather bright major third of approximately 407.8¢. However, a 9/8 plus the minor whole tone [[10/9]] yields [[5/4]]. This distinction, between a major whole tone and minor whole tone, has been completely obliterated in [[12edo]], and so we are unaccustomed to thinking of more than one size of whole tone comprising a major third. Other systems that temper out this difference (which is [[81/80]], the syntonic comma of about 21.5¢), such as [[19edo]], [[26edo]], and [[31edo]], are called [[meantone]] temperaments. | Two 9/8's stacked produce [[81/64]], the Pythagorean major third, a rather bright major third of approximately 407.8¢. However, a 9/8 plus the minor whole tone [[10/9]] yields [[5/4]]. This distinction, between a major whole tone and minor whole tone, has been completely obliterated in [[12edo]], and so we are unaccustomed to thinking of more than one size of whole tone comprising a major third. Other systems that temper out this difference (which is [[81/80]], the syntonic comma of about 21.5¢), such as [[19edo]], [[26edo]], and [[31edo]], are called [[meantone]] temperaments. | ||
A stack of six intervals of 9/8 exceeds the octave by the [[Pythagorean comma]]. | |||
== History == | == History == | ||
The (whole) tone as an interval measure was already known in Ancient Greece. | The (whole) tone as an interval measure was already known in Ancient Greece. {{w|Aristoxenus}} (fl. 335 BC) defined the tone as the difference between the [[3/2|just fifth (3/2)]] and the [[4/3|just fourth (4/3)]]. From this base size, he derived the size of other intervals as multiples or fractions of the tone, so for instance the just fourth was 2½ tones in size, which implies [[12edo]]. | ||
== Temperaments == | == Temperaments == | ||
In [[meantone]], 9/8 is equated with [[10/9]], so that two instances of 9/8~10/9 stack to ~[[5/4]]. [[Superpyth]] instead sharpens 9/8 to equate it with [[8/7]]. | |||
Since 9/8 is reached by stacking two instances of [[3/2]], temperaments in subgroups that include 3 cannot be generated by ~9/8. However, it can be a generator in subgroups such as [[2.9.5.7 subgroup|2.9.5.7]], where it generates [[Subgroup temperaments #Baldy|baldy]] for example. | |||
== Approximation == | |||
9/8 is well-represented in [[6edo]] and its multiples, though only multiples of [[12edo]] (up to [[300edo]]) map 9/8 to 1\6 by [[patent val]]. [[Edo]]s which tune [[3/2]] close to just, such as [[29edo]], [[41edo]], and [[53edo]], will tune 9/8 close to just as well. | |||
{{Interval edo approximation|9/8}} | |||
== Notation == | |||
In musical notations that employ the [[5L 2s|diatonic]] [[chain-of-fifths notation|chain-of-fifths]], such as the [[ups and downs notation]], the whole tone is represented by the distances between the notes A–B, C–D, D–E, F–G, and G–A. | |||
The scale is structured with the following step pattern: | |||
* A to B: [[9/8|whole tone]] | |||
* B to C: [[256/243|limma]] | |||
* C to D: [[9/8|whole tone]] | |||
* D to E: [[9/8|whole tone]] | |||
* E to F: [[256/243|limma]] | |||
* F to G: [[9/8|whole tone]] | |||
* G to A: [[9/8|whole tone]] | |||
This pattern highlights the placement of the whole tone intervals between the note pairs above, distinguishing them from the [[256/243|limma]] that occurs between the other note pairs. | |||
== See also == | == See also == | ||
| Line 31: | Line 51: | ||
[[Category:Second]] | [[Category:Second]] | ||
[[Category:Whole tone]] | [[Category:Whole tone]] | ||
[[Category:Greek music]] | [[Category:Ancient Greek music]] | ||
[[Category:Commas named after their interval size]] | |||
Latest revision as of 04:25, 12 March 2026
| Interval information |
Pythagorean major second
reduced,
reduced harmonic
[sound info]
9/8, the Pythagorean whole tone or major second, is an interval measuring approximately 203.9¢. It can be arrived at by stacking two just perfect fifths (3/2) and reducing the result by one octave. However, it is also a relatively low overtone in its own right, octave-reduced. It can be treated as a dissonance or a consonance, depending on compositional context, though because of its relatively close proximity to the unison, it is the largest superparticular interval known to cause crowding, which lends more to it being considered a type of dissonance- at least in historical Western Classical traditions and in the xenharmonic traditions derived from them.
Two 9/8's stacked produce 81/64, the Pythagorean major third, a rather bright major third of approximately 407.8¢. However, a 9/8 plus the minor whole tone 10/9 yields 5/4. This distinction, between a major whole tone and minor whole tone, has been completely obliterated in 12edo, and so we are unaccustomed to thinking of more than one size of whole tone comprising a major third. Other systems that temper out this difference (which is 81/80, the syntonic comma of about 21.5¢), such as 19edo, 26edo, and 31edo, are called meantone temperaments.
A stack of six intervals of 9/8 exceeds the octave by the Pythagorean comma.
History
The (whole) tone as an interval measure was already known in Ancient Greece. Aristoxenus (fl. 335 BC) defined the tone as the difference between the just fifth (3/2) and the just fourth (4/3). From this base size, he derived the size of other intervals as multiples or fractions of the tone, so for instance the just fourth was 2½ tones in size, which implies 12edo.
Temperaments
In meantone, 9/8 is equated with 10/9, so that two instances of 9/8~10/9 stack to ~5/4. Superpyth instead sharpens 9/8 to equate it with 8/7.
Since 9/8 is reached by stacking two instances of 3/2, temperaments in subgroups that include 3 cannot be generated by ~9/8. However, it can be a generator in subgroups such as 2.9.5.7, where it generates baldy for example.
Approximation
9/8 is well-represented in 6edo and its multiples, though only multiples of 12edo (up to 300edo) map 9/8 to 1\6 by patent val. Edos which tune 3/2 close to just, such as 29edo, 41edo, and 53edo, will tune 9/8 close to just as well.
| Edo | Step size | Cents (¢) | Absolute error (¢) | Relative error (%) |
|---|---|---|---|---|
| 6 | 1\6 | 200.00 | -3.91 | -1.96 |
| 12 | 2\12 | 200.00 | -3.91 | -3.91 |
| 18 | 3\18 | 200.00 | -3.91 | -5.87 |
| 23 | 4\23 | 208.70 | +4.79 | +9.17 |
| 24 | 4\24 | 200.00 | -3.91 | -7.82 |
| 29 | 5\29 | 206.90 | +2.99 | +7.22 |
| 30 | 5\30 | 200.00 | -3.91 | -9.78 |
| 35 | 6\35 | 205.71 | +1.80 | +5.26 |
| 41 | 7\41 | 204.88 | +0.97 | +3.31 |
| 47 | 8\47 | 204.26 | +0.35 | +1.35 |
| 53 | 9\53 | 203.77 | -0.14 | -0.60 |
| 59 | 10\59 | 203.39 | -0.52 | -2.56 |
| 65 | 11\65 | 203.08 | -0.83 | -4.51 |
| 71 | 12\71 | 202.82 | -1.09 | -6.47 |
| 76 | 13\76 | 205.26 | +1.35 | +8.57 |
| 77 | 13\77 | 202.60 | -1.31 | -8.42 |
Notation
In musical notations that employ the diatonic chain-of-fifths, such as the ups and downs notation, the whole tone is represented by the distances between the notes A–B, C–D, D–E, F–G, and G–A.
The scale is structured with the following step pattern:
- A to B: whole tone
- B to C: limma
- C to D: whole tone
- D to E: whole tone
- E to F: limma
- F to G: whole tone
- G to A: whole tone
This pattern highlights the placement of the whole tone intervals between the note pairs above, distinguishing them from the limma that occurs between the other note pairs.
See also
- 16/9 – its octave complement
- 4/3 – its fifth complement
- 32/27 – its fourth complement
- Gallery of just intervals
- List of superparticular intervals