Microtonal, just intonation, electronic music software Microtonal, just intonation, electronic music software

Encyclopedia of Microtonal Music Theory

@ 00 A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
Login   |  Encyclopedia Index

wolf

[from Margo Schulter, posting to Tuning Digest # 1597]

An "unusual" [musical] interval in a given tuning system and style which is felt not to be freely substitutable for an "expected" variety of the same interval.

Sometimes it is asserted that "Wolf" intervals are "unusable" or "unplayable," but this judgment is both contextual and often partial: there are often specific sonorities and usages where these intervals are musically useful even in styles where they are generally considered to be "too out-of-tune" for most purposes.

. . . . . . . . .
[Joe Monzo]

Wolf intervals are a charactistic feature of most tunings belonging to the meantone family of temperaments; in particular, there is a "wolf 5th" which occurs at the interval which connects the last and first notes in the meantone chain-of-5ths; this is illustrated below.

Meantone tunings most typically have 12 notes, and the most typical set of notes in a meantone chain is:

Eb -- Bb -- F -- C -- G -- D -- A -- E -- B -- F# -- C# -- G#

So the wolf occurs between G#:Eb.

The exact size of the wolf depends on the amount of tempering used to narrow the meantone generator "5th", the size of the wolf widening in direct proportion to the amount of tempering. Here are some examples which are historically and/or theoretically important (along with their ET approximations):

  meantone                                         ~cents difference  logarithmic ratio
  tempering    ~cents of wolf  ~cents of generator  wolf - generator   wolf:generator    ~= EDO

- 1/3-comma      757.3513857       694.7862377        62.565148         1.090049492       19-ET
- 2/7-comma      746.0861864       695.8103467        50.27583966       1.072255091       50-ET
- 5/18-comma     744.2086531       695.9810315        48.2276216        1.069294448       50-ET
- 7/26-comma     742.1866943       696.164846         46.02184831       1.066107688       50-ET
- 1/4-comma      737.6372869       696.5784285        41.05885841       1.058943626       31-ET
- 2/9-comma      731.0659206       697.1758254        33.89009521       1.048610543      105-ET
- 3/14-comma     729.1883874       697.3465102        31.84187715       1.045661485       74-ET
- 1/5-comma      725.8088276       697.6537429        28.15508465       1.040356817       43-ET
- 1/6-comma      717.9231881       698.3706193        19.55256881       1.02799741        55-ET
- 1/11-comma     700.0012801       699.9998836         0.001396448      1.000001995       12-ET
		

Note that the wolf-"5th" disappears in 12-edo tuning, which is essentially the same as 1/11-comma meantone.

. . . . . . . . .

The tonalsoft.com website is almost entirely the work of one person: me, Joe Monzo. Please reward me for my knowledge and effort by choosing your preferred level of financial support. Thank you.

support level