
© 2004 Tonalsoft Inc. 
devised by George Secor, and Dave Keenan and Paul Erlich
©2001 by Joe Monzo
In April 2001, Paul Erlich and Dave Keenan were responding to a request from Joseph Pehrson, on the Yahoo Tuning List, for good 19tone 11limit subsets of 72EDO.
Dave realised that, although there was no obvious winner, most of the good ones were subsets of a particular 31 tone periodicity block which was very even melodically. It was a 72EDO tempering of a 31tone planar microtemperament that he had created in December 1999 in collaboration with Paul Erlich and Carl Lumma.
Dave posted this 31tone scale to the tuning list, whereupon Paul Erlich pointed out that:
Paul suggested that Blackjack would be an excellent name for the latter.
Then Dave Keenan performed a computerassisted search for other singlechain 11limit generators which either gave no fewer hexads per note at greater accuracy, or gave more 11limit quasijust hexads per note. He repeated the search at the 7limit (for tetrads instead of hexads). There were none. In fact no other generator even came close.
This generator divides the 12EDO "5th" into 6 logarithmically equal parts or steps, whereas it is usually divided into 7. In mathematical language this is described as (2^{(7/12)})^{(1/6)} [= 116&2/3 cents]. Thus it also divides the "neutral 3rd" into 3 equal parts. (See Graham Breed's neutral 3rd lattices.)
Any tuning with a generator in the range between 116.1 cents (= 31EDO) and 117.8 cents (= 0.7 cents larger than 41EDO) has MIRACLE properties, but that of 72EDO comes closest to the optimal generators calculated by a number of methods (including root mean square and maximal absolute). There is no single optimum MIRACLE generator; many different kinds of optimum are within 0.15 cent either side of 72EDO.
(At the time of its rediscovery in April 2001, Dave, Paul, and Carl were all unaware that George Secor had published this generator  but not the MOS scales  in 1975, in his article A New Look at the Partch Monophonic Fabric, originally appearing in Xenharmonikôn 3. In honor of George, this interval was subsequently named the secor.)
Graham Breed has pointed out that 10 and 11EDO, whose generators lie a bit outside this range, exhibit some of the same melodic properties as the MIRACLE scales. These facts:
have led Graham to advocate the use of a decimalbased notational system for scales in the MIRACLE family. (See also Graham's decimal page.)
The Blackjack generator is also nearly identical to an interval the size of 7 degrees of 72EDO [= exactly 116 & 2/3 cents, or 1 & 1/6 Semitones]. Thus, it can be represented quite accurately as a 21outof 72EDO tuning. Because of the many useful properties associated with 72EDO, this greatly simplifies many aspects of the presentation of the Blackjack scale, such as its notation or its diagramming on a lattice.
I will present the Blackjack scale here as 21outof72EDO. In the 72edo version of blackjack, L is 5 degrees of 72edo and s is 2 degree, so L = 2.5s.
It was decided by Dave and others that "D" should be the reference note for the system, because the layout of notes on the standard Halberstadt keyboard is symmetrical around "D". But I used "C" as the reference in my examples here.
Here is a graph showing the pitchheight of the notes in this scale, within one "octave". Each note is labeled with both my ASCII adaptation of the Sims/Herf 72EDO notation, and its Semitone value.
Below is a 4line staff notation I developed for Blackjack, based on Graham Breed's decimal symbology for the MIRACLE family of temperaments. The centsvalue is given above the staff, and the 72EDO degree below.
The reference pitch ("0") occurs on the ledger line between staves, and each successive ascending space and line represents the next secorsized positive generator in the decimal series, and thus the next cardinality in Breed's notation: 1, 2, 3, ... 9, and when that set is exhausted, the next (and last) in the series of generators, 0v, is notated as the "0" reference pitch, an "8ve" higher, accompanied by the v symbol which indicates lowering by a quomma. Descending from the higher "0" ledgerline is the negative series of generators, 9^, 8^, 7^, ... 0^, notated on the same line or space as the namesake cardinality, but accompanied by a ^ symbol to indicate raising by a quomma.
Below is a 5limit bingocardlattice of 72edo, with the blackjack scale shown in buff in the central periodicityblock and in pink in the blocks which are commatic equivalents.
Here is an
interval
matrix chart of all
dyadic
intervals available in the 21outof72 Blackjack scale,
with the pitches labeled with
their 21tone Blackjack degree numbers and Semitone values.
All interval sizes are shown in Semitones.
(Thanks to Paul Erlich for colorcoding the intervals according to the odd limits of the JI intervals they represent.)
Paul Erlich made a nifty colorcoded chart of blackjack intervals (... you have to be a member of Yahoo groups to view it).
Paul also made a 7limit lattice diagram showing the periodicity blocks implied by the Blackjack tuning, as well as illustrating the many harmonic structures implied by this scale. I have adapted it here to my own ASCII 72EDO notation. The Blackjack scale is wafsojust with respect to this lattice.
Here is a mapping which I designed, placing 72EDO onto the fingerboard/keyboard of a Starr Labs Ztar instrument, showing all 72EDO degrees and their ASCII Monzo notation. The placement of black and white keys reflects the association of various 72EDO notes with those in 12EDO as they appear on a regular Halberstadt piano keyboard. The Blackjack notes are shown in orange.
Notice how the placement of the Blackjack notes in the above mapping shifts upward by one key as one travels to the right, because the generator of 7/72 is one more than the 6/72 steps in each column of the keymap.
In the mapping below, I adapted the Starr Labs Zboard keyboard so that each column is 7 steps high, thereby making all the Blackjack notes adjacent.
Graham Breed presented a "comma pump" chord progression in the Blackjack tuning (click on graphic to hear mp3, looped 3 times):
Below is a score of Graham's progression in my 4linestaff adaptation of Graham's
decimal notation:
Below is a score of the progression notated in my
72edo
HEWM notation:
Below is an applet which shows the 5limit representation
of Graham's chord progression. Mouseover the chordnumber
(without clicking) to
see a lattice of that chord in red. Commatic equivalents are
shown in purple. Note that Graham's chords all imply a 7limit
harmony, which is shown in its closest 5limit approximation (225:128
above the "root" of the chord) here.
Graham's chords are as follows:
For the benefit of those wishing to map this family of tunings to a standard 12tone Halberstadt keyboard, Paul Erlich devised an interesting 12tone subset of Blackjack, presented in Tuning List post 22532 from Sat May 12, 2001 8:35 am.
Dave Keenan gave an analysis of it in Tuning List post 22622 from Sun May 13, 2001 6:02 am.
Dave also created this diagram of a colorcoded design for mapping the full Blackjack scale to the Halberstadt keyboard.
As stated above, 31edo is the tuning which provides the lower limit of the blackjack generator, so those who work in 31edo may easily form blackjack as a subset of that tuning. Below is a table showing the degrees of 31edo which form blackjack:
Below are two pitchheight graphs showing the 31edo version of blackjack. The graph on the left has the "octave" divided into 12 steps, and that on the right has it divided into 31 steps.
In the 31edo version of blackjack, L is 2 degrees of 31edo and s is 1 degree, so L = 2s.
Below is a 5limit bingocardlattice of 31edo, with the 21tone blackjack scale shown in buff in the central part of the blackjack chain which passes thru n^{0} and in pink in the chains which are commatic equivalents. The central periodicityblock contains most of the blackjack scale, with 7 of the notes falling into commaticallyequivalent chains. The general southwesttonortheast trend of blackjack is obvious; compare to the 72edo lattice above.
Below are two pitchheight graphs showing the 41edo version of blackjack. The graph on the left has the "octave" divided into 12 steps, and that on the right has it divided into 41 steps.
In the 41edo version of blackjack, L is 3 degrees of 41edo and s is 1 degree, so L = 3s.
Links to audiofiles of pieces composed in blackjack:
(search the Yahoo Tuning List archives for further background info)
updated: