Pythagorean comma: Difference between revisions
m - typo. Wording, style and links |
This is not a negative interval! Diatonic degrees don't take positive or negative cuz they are ordinals. Pitch does. Also remove "tunings close to [Pythagorean tuning]" as that's vague |
||
| (6 intermediate revisions by 4 users not shown) | |||
| Line 1: | Line 1: | ||
{{ | {{Interwiki | ||
| en = Pythagorean comma | |||
| de = 531441/524288 | | de = 531441/524288 | ||
| es = | | es = | ||
| ja = | | ja = | ||
| Line 8: | Line 8: | ||
| Ratio = 531441/524288 | | Ratio = 531441/524288 | ||
| Name = Pythagorean comma, ditonic comma | | Name = Pythagorean comma, ditonic comma | ||
| Color name = | | Color name = LLwM, lalawama<br>pM, poma | ||
| Comma = yes | | Comma = yes | ||
}} | }} | ||
{{Wikipedia| Pythagorean comma }} | {{Wikipedia| Pythagorean comma }} | ||
The '''Pythagorean comma''' or '''ditonic comma''' is the interval with the ratio '''531441/524288''' ([[monzo]]: {{monzo| -19 12 }}). It is the amount by which twelve fifths exceed seven octaves, or in other words (3/2)<sup>12</sup>/2<sup>7</sup>. It also can be written as the ratio between the apotome and limma, ([[2187/2048]])/([[256/243]]), and as the ratio between the Pythagorean augmented fourth and the Pythagorean diminished fifth, ([[729/512]])/([[1024/729]]). In addition, it is also the difference | The '''Pythagorean comma''' or '''ditonic comma''' is the interval with the ratio '''531441/524288''' ([[monzo]]: {{monzo| -19 12 }}). It is the amount by which twelve [[3/2|fifths]] exceed seven [[2/1|octaves]], or in other words (3/2)<sup>12</sup>/2<sup>7</sup>. It also can be written as the ratio between the Pythagorean apotome and limma, ([[2187/2048]])/([[256/243]]), and as the ratio between the Pythagorean augmented fourth and the Pythagorean diminished fifth, ([[729/512]])/([[1024/729]]). In addition, it is also the difference between six [[9/8]] major seconds (i.e. an augmented seventh) and an octave. | ||
In [[ | In [[Pythagorean tuning]], this interval is an ''inverse'' diminished second. This is because adding Pythagorean commas makes the interval go up in pitch, down in [[5L 2s|diatonic]] [[degree]]s. This apparently counterintuitive notion is a result of just fifths naturally producing a [[TAMNAMS #Step ratio spectrum|hard-of-basic]] diatonic scale, which means that the [[chromatic semitone]] is wider, not narrower, than the [[diatonic semitone]]. | ||
== Temperaments == | == Temperaments == | ||
If the | If the Pythagorean comma is [[tempering out|tempered out]], then the [[circle of fifths]] closes at 12 notes. This circle of fifths covers the entirety of [[12edo]], while larger multiples of 12edo such as [[24edo]] and [[72edo]] contain multiple such circles. If one takes this circle of fifths and adds an independent [[generator]] for prime [[5/1|5]], this leads to the 5-limit rank-2 [[compton]] temperament. See [[Compton family]] for the family of rank-2 temperaments where it is tempered out. | ||
Edos with a fifth sharper than the 12edo fifth of 700{{c}}, such as [[41edo]] and [[53edo]], map the Pythagorean comma to a positive small number of steps rather than tempering it out. The Pythagorean comma is quite close to the [[81/80|syntonic comma]], only exceeding it by a [[schisma]]. It is also fairly close to the [[64/63|septimal comma]], with the septimal comma exceeding the Pythagorean comma by the [[garischisma]]. Tempering out both the schisma and the garischisma leads to [[garibaldi]] temperament, which is one of the most intuitive [[7-limit]] interpretations of the Pythagorean chain of fifths. | |||
Edos with a fifth | Edos with a fifth flatter than the 12edo fifth, such as [[19edo]] and [[31edo]], map the Pythagorean comma negatively, and thus have a positive diminished second (also known as a [[diesis (scale theory)|diesis]]). The majority of these edos support [[meantone]], which equates the Pythagorean major third [[81/64]] to the 5-limit major third [[5/4]]. | ||
Since it is reached by 12 fifths, a highly composite number, there are many temperaments that split this comma whilst keeping fifths unsplit, splitting octaves instead. Notably: | |||
* [[Kalismic]], splitting it into 2 [[2835/2816|fwiwismas]]. | |||
* [[Landscape]], splitting it into 3 [[225/224|marvel commas]]. | |||
* [[Nexus]], splitting it into 3 [[243/242|rastmas]]. | |||
* [[Atomic]], splitting it into 12 schismas. | |||
== See also == | == See also == | ||
* [[Mercator's comma]], the difference between 53 perfect fifths and 31 octaves | * [[Mercator's comma]], the difference between 53 perfect fifths and 31 octaves | ||
* [[41-comma]], the difference between 65 octaves and 41 perfect fifths | |||
* [[Gallery of just intervals]] | * [[Gallery of just intervals]] | ||
* [[Small comma]] | * [[Small comma]] | ||