Kraig Grady's 12 tone 7-limit scale and extensions showing other related scales developed by Wilson, Narushima, Poole, Schulter, Heathwaite, and White.
This chart shows the harmonic relations found in this tuning
A cross section showing each note compared to every other
A chart of all the intervals found and the variations found of each basic scale step size.
Here are the frequencies in case someone desires to use in the original key or make a compatible instrument.
This is a chart of the 41 simple tetrads found in centaur.
This chart shows the various diatonic like scale scales expressed in a cycle of fifths. It melodic nature of this series help in determining some of the pitches chosen over others.
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 returns to the beginning.
It has been pointed out by Praveen Venkataramana that one can combine any of the like pairs in what he calls a Reverse Viggo Brun method with the result forming a constant structure.
this PDF shows the possible 10 through 6 tone scales that can be found using this method. Centaur Constant Structures
This document has quite a few 12 tone 7 limit tunings
Here also is the 7 Limit used by Other Music as well as an option that was the strongest option when i first was making Centaur. Its advantage is another subharmonic series of 1-3-7-9 and was one that I had a hard time deciding between it and the present centaur. That Centaur provides another 1-3-5 along with repeated tetrachords made me decide in its favor but its own benefits are not to be ignored. Centaur can also be said that there is no tone that doesn't have a 3/2 above or below it. leaving no tone somewhat suspended.
An interesting 15 tone 11 limit extension by Praveen Venkataramana being a somewhat rare numbered Constant Structures.
Other related lattices can be found on his page venkataramana.html
Here are two from Terumi Narushima showing 17 and 19 tone constant structure scales containing Centaur.
SOME CLOSE RELATIVES TO 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.
This is a 17 tone scale of Wilson has a 10 tone overlap with Centaur and extends into the 11 limit territory with two subharmonic hexads instead of only one. Conceptually it shares the idea of having both harmonic and/or 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.
Here is a 19 tone extension of Centaur.
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
INDEPENDENT SCALES HAVING CLOSE RELATIONS WITH CENTAUR
Monarda is a JI version of Scott Dakota's scale that explores 2 harmonic sets a 3/2 apart, but exploring the use of the 17th harmonic.
Monarda inspired the slight variation below by myself which exchanges some features at the expense of others.
Justin White devised the following scale after seeing the19 tone 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 14/9, 21/20 were left out because they are too close to 25/16 and 25/24)"
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.
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.
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.
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.
The whole process is interesting in showing how a just intonation scale can be taken as a starting point, greatly expanded or contracted into something related. Lastly how similar Just intonation scales can be developed independently and have many points of overlap and shared contact.