Kraig Grady's 12 tone 7-limit scale and extensions showing other related scales developed by Wilson, Narushima, Poole, Schulter, Heathwaite, and White.






This diagram below was found in one of Wilson's notebooks.

The focus is on a 10-tone constant structure subset similar to the scales generated by the Viggo Brun algorithm.

This diagram illustrates a subset of 7-tone scales contained therein, showing the 10 cycles before it reterns to the beginning.

It has been pointed out that one can combine any of the like pairs with the result forming a Constant structure.

this PDF shows the possible 10 through 6 tone scales that can be found in such fashion. Centaur Constant Structures

Here are the frequencies in case someone desires to make a compatible instrument.

centaur pitches


Centaur Extensions - 17 &19 Tone Scales

Here are two from Terumi Narushima showing 17 and 19 tone constant structure scales containing Centaur.






This is a scale of Wilson which extends the 7 limit territory of Centaur to an 11 limit one with two subharmonic hexads instead of only one. While it differs from containing Centaur with two notes, it captures the idea of havinging both harmonic and subharmonic forms a 3/2 apart.

The scale is discussed on this page of the Gems of the Wilson Archives Blog


Independently this 12 tone subset of the above scale was found looking for a Centaur-like scale with 11s instead of 5s.

sisuitl scale

sisuitl cycle

Here is a 19 tone extension of Centaur.





This is a very closely related 14-tone scale that shares 11 notes in common with the Centaur scale in two instances a fifth apart

It first appeared in Lou Harrison's music primer which excerpt I include:





and here is Wilson's 19 tone scale mentioned which one can compare to Terumi Narushima's 19 tone scale above

where a 14/9 and 5/3 are uses instead of a 27/16 and 63/40



This still another solution made by Wilson uses the 35/27 and 35/18 instead of the 21/16 and 63/32 of the Narushima above.



Below is a slight variation worked out especially when writing for string ensembles where one might wish to preserve the traditional open string tunings.

A discussion on it can be found here http://anaphoriasouth.blogspot.com.au/2014/01/a-19-tone-extension-of-centaur-for.html



Justin White devised the following scale after seeing the one above that also is an epimoric scale. White Described it as, "Designed to incorporate more resources from the Boomsliter and Creel Blues reference, but omitting notes that are too close to already existing pitches (e.g 75/64, 225/128 were left out because they are too close to 7/6 and 7/4)"

justinwhite's 19 tone 7 limit scale

An interesting 15 tone 11 limit extension by Praveen Venkataramana being a somewhat rare numbered Constant Structures.

15 tone centaur extension




The guitarist and improviser Rod Poole originally used Centaur as the base pitches to construct his various tunings on his guitar. While over time the variations took him in a new direction, it shows how a scale can develop into new territory.

Here is one suggestion Erv Wilson made to Rod that includes Centaur within a 22-tone scale.



Here is a 29 tone scale Erv attributes to Rod Poole three months later than the above. This scale adds 17 tones to Centaur. It is worth noting that the 17 that Rod ended up using is marked with small check marks on close examination. Margo Schulter has written here about another of their scale interactions. Margo Schulter has developedan extended array of related scales to Centaur and Poole's scale that warranted her own page in The Main Library.

poole's 29 tone scale


This is the last scale we have of Rod Poole. Centaur is in red of which the hollow notes were omitted as he decided to drop the 5 limit ratios and expand the scale in both the 11 and 13 direction.

pool's 17


Margo Schulter has developed quite a few related scales to Centaur and Poole's scale that can be found on her page in The Main Library.


Andrew Heathwaite's guitar tuning Cartwheel is another17 tone 13 limit constant structure that overlaps significantly with both Centaur, Poole's, and Schulter's tunings focusing on the harmonic direction.

cartwheel scale


Here is a closely related scale by Owen Gaertner of the band 'Horse Lords' who like the other guitar tunings can use various open tunings. In his case he seems to be quite specific in using an open tuning of 6/8/12/16/21/24.

gaertner's 13 limit scale


The whole process is interesting in showing how a just intonation scale was taken as a starting point, greatly expanded and then contracted into something related, but a work of its own.


main page