3/2: Difference between revisions
ArrowHead294 (talk | contribs) |
→In regular temperament theory: I'll do you one better |
||
| (7 intermediate revisions by 3 users not shown) | |||
| Line 14: | Line 14: | ||
{{Wikipedia|Perfect fifth}} | {{Wikipedia|Perfect fifth}} | ||
'''3/2''', the '''just perfect fifth''', is a very [[consonance|consonant]] interval, due to the numerator and denominator of its ratio being very small numbers. Only the [[2/1|octave]] and the [[3/1| | '''3/2''', the '''just perfect fifth''', is a very [[consonance|consonant]] interval, due to the numerator and denominator of its ratio being very small numbers. Only the [[2/1|octave]] and the [[3/1|tritave]] have smaller numbers. | ||
== Properties == | == Properties == | ||
| Line 27: | Line 27: | ||
=== In regular temperament theory === | === In regular temperament theory === | ||
Because 3/2 | Because 3/2 is a very simple and concordant interval, it is still recognizable even when heavily tempered. Often it is tempered so that an octave-reduced stack of fourths or fifths approximates some other interval. Some examples: | ||
[[Meantone]] temperament flattens the fifth from just such that the major third generated by stacking four fifths is closer to (or even identical to) 5/4. The minor 3rd generated by stacking three fourths is closer to 6/5. | [[Meantone]] temperament flattens the fifth from just (to around 695 cents) such that the major third generated by stacking four fifths is closer to (or even identical to) 5/4. The minor 3rd generated by stacking three fourths is closer to 6/5. | ||
[[Superpyth]] temperaments ''sharpen'' the fifth from just so that the major third is closer to 9/7 and the minor third is closer to 7/6. Thus the minor 7th 16/9 approximates 7/4 instead of 9/5. | [[Superpyth]] temperaments ''sharpen'' the fifth from just so that the major third is closer to 9/7 and the minor third is closer to 7/6. Thus the minor 7th 16/9 approximates 7/4 instead of 9/5. | ||
[[Schismatic]] temperament | * One may choose to prioritize the accurate tuning of either the thirds or the harmonic seventh, leading to a ~710c tuning when prioritizing the thirds, or a ~715c tuning when prioritizing 7/4. | ||
[[Schismatic|Schismic]] temperament adjusts the fifth such that the ''diminished'' fourth generated by stacking eight fourths approximates 5/4. As this is already a close approximation, the tuning of the fifth can be varied around its just tuning, but is most simply flattened by a tiny amount. Thus a triad with 5/4 is written as {{nowrap|{{dash|C, F♭, G}}}} (unless the notation has accidentals for [[81/80]], e.g. {{nowrap|{{dash|C, vE, G}}}}). | |||
* Garibaldi temperament is an extension of schismatic that sharpens the fifth so that the small interval between the major third and diminished fourth can also be used to create simple 7-limit intervals. | |||
== Approximations by edos == | == Approximations by edos == | ||
| Line 43: | Line 47: | ||
|- | |- | ||
! [[Edo]] | ! [[Edo]] | ||
! class="unsortable" | | ! class="unsortable" | Deg\edo | ||
! Absolute<br> | ! Absolute <br>error ([[Cent|¢]]) | ||
! Relative | ! Relative <br>error (%) | ||
! ↕ | ! ↕ | ||
! class="unsortable" | Equally accurate | ! class="unsortable" | Equally accurate <br>multiples | ||
multiples | |||
|- | |- | ||
| [[12edo|12]] || 7\12 || 1.955 || 1.955 || ↓ || [[24edo|14\24]], [[36edo|21\36]] | | [[12edo|12]] || 7\12 || 1.955 || 1.955 || ↓ || [[24edo|14\24]], [[36edo|21\36]] | ||
| Line 251: | Line 253: | ||
|- | |- | ||
| [[30edo]] | | [[30edo]] | ||
| | | 18\30 | ||
| 720.000 | | 720.000 | ||
| pentatonic edo | | pentatonic edo | ||