For example, in the syntonic temperament (which tempers out 81/80), all JI ... draw a unison vector (which connects two notes that are a comma apart), and a line ...
The Tone Diamond As you can explore here,1 we have implemented a mapping of the Thummer keyboard’s note-‐ layout to the QWERTY keyboard which allows (a) Dynamic Tuning along the syntonic2 tuning continuum, (b) the dynamic alignment of notes’ fundamentals & overtones with the tunings’ scale steps to maximize consonance, and (c) the ability to adjust overtones from fully harmonic to fully tempered in any tuning. Question: Can there be a fundamental tempering slider that can take us from a JI tuning to a fully tempered tuning such that the geometric shape of a given musical structure stays the same whether tempered, JI, or in between (and which uses a sensible algorithm to reverse-‐ map from temperament to JI)? Yes we can. Question: Can we progress smoothly from JI to regular temperament maintaining maximal consonance all the way (thus creating a new trade-‐off between regularity and harmonicity)? Yes we can. Question: Does this enable us to propose a new type of diamond control system which allows for the user to easily control the trade-‐offs between the affects of regularity, harmonicity, and consonance? Of course!
1. Reverse-‐mapping from regular temperament to Just intonation In a regular temperament, a JI comma is tempered to unison. This means that a given note in a regular temperament has an infinity of JI notes mapped to it—all of which are comma-‐variants of one another. For example, in the syntonic temperament (which tempers out 81/80), all JI notes of the form
(where z is an integer)—such as 100/81, 5/4, 81/64—are all mapped to
the same note in the temperament. When mapping in reverse from a regular temperament to JI there is no mathematical reason to choose any one of these comma-‐variants over another. But there are sensible aesthetic choices. Let’s look at some possible reverse-‐mappings of the syntonic diatonic scale C, D, E, F, G, A, B to JI: Table 1. Three valid reverse-‐mappings of the syntonic diatonic scale.
C 1/1 1/1 1/1
D 9/8 10/9 9/8
E 5/4 5/4 81/64
F 4/3 4/3 4/3
G 3/2 3/2 3/2
A 5/3 5/3 81/48
B 15/8 15/8 15/8
1
http://www.thummer.com/blog/2007/06/dynamic-‐tuning-‐mark-‐i.html
2
Syntonic tuning continuum: http://www.thummer.com/blog/2007/06/isomorphic-‐controllers-‐and-‐ dynamic.html
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The above are all legitimate reverse-‐mappings from rank-‐2 regular temperament to JI, but the first two contain five JI major and minor triads, whereas the third contains only one JI triad (the roots of JI major triads are coloured blue, minor are coloured pink). The first two reverse-‐ mappings are special because there is no other reverse-‐mapping (other than a simple transposition) that can provide as many JI triads as the first two. It makes aesthetic sense to reverse-‐map to a choice of JI notes that provides for the densest clustering of major and minor triads; it also makes sense for that clustering of JI triads to be centred around the Thummer’s reference tone (the 1/1), which on the Thummer is (by default) the note D. The task is, therefore, to determine a method to reverse-‐map from any temperament to JI such that we achieve the densest possible clustering of JI triads centred around the reference tone.
2. Method to reverse-‐map to a triad-‐dense JI block centred over the reference tone This section may be a bit heavy so, if necessary, skip it and go straight to Section 3. If we draw a JI lattice with unit vectors of 3/2 and 5/4, we get the tightest possible spatial clustering of JI major and minor triads (3/2 and 6/5 would have been an alternative valid choice). A different choice of 5-‐limit JI basis for the unit vectors results in a less dense clustering of JI triads—as shown below where the notes making up all triads containing the note C are coloured green for JI lattices with two different unit vectors.
Figure 2. All triads containing C in JI lattice basis (unit vectors) of 3/2 and 5/4.
Figure 1. All triads containing C in JI lattice basis (unit vectors) of 5/3 and 25/24.
This implies that, for any reverse-‐mapping to a selection of JI notes to have as many JI triads in it as possible, the selection must form a block from the lattice with unit vectors of 3/2 and 5/4. Mathematically, we cannot reverse-‐map to a JI note and one or more of that note’s comma-‐ variants because each different tempered note maps to a different set of comma-‐variants. If we draw a unison vector (which connects two notes that are a comma apart), and a line at right angles to the unison vector through each of that unison vector’s end points, the strip The Tone Diamond
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(coloured green below) within these lines (such that one line is included and one is not) contains no notes that differ by a comma (by containing a note I mean containing the centre point of each square). Any block taken from this strip, therefore, contains the notes that can be reverse-‐mapped to and also have the densest possible clustering of JI major and minor triads. This is illustrated below for the syntonic comma—the two Ds (and Fbbs and B##s) are hashed because either one (but not both) can be included in the green strip—depending on which line is included in the green strip.
Figure 3. Syntonic strip.
Within this strip is the tightest possible clustering of major and minor triads for the syntonic temperament; but at the edges of the strip, JI triads are no longer possible—e.g. there are no JI Eb, C##, or Cb major triads within the strip. Because the breakdown occurs at the edges of the strip we need to ensure that the strip edges are located as far away from the regular temperament’s reference tone (D) as is possible.
Centring the strip It is necessary to place the strip so that the just chords are centred on the button-‐lattice’s layout around the reference tone (D). This is the tricky bit…not all of it is fully worked out…essentially in all the examples I’ve tested so far, the strip needs to be placed so that one of its edges runs through the reference tone (D). This is what I have done in all the figures in this write-‐up.
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I have a rough proof for why this works when the temperament has a generator which is a direct mapping of a primary consonance (e.g. syntonic, magic, hanson, schismatic, etc.). I think it will work for all temperaments, but I don’t have the general proof yet. For now we will assume that the strip must be located so that its edge runs through the reference tone (D).
Delimiting the strip So we have a strip—the first part of this section told us how wide it needed to be; the second part gave us the precise location of the strip on the JI lattice; the final task is to delimit the strip into the block that is mapped to the Thummer’s buttons. At any given tuning (where
), the Thummer has a finite number
of -‐reduced
notes. The width of the block is determined by the unison vector; for example, if the comma is
, its unison vector length is
and its width is
. So if the block needs to contain
, its length must be
notes
(this length being inclusive of the lines),
and this block is centred over the reference note (D).
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Two such blocks are shown below—syntonic and Magic, and
is here assumed to be 19:
Figure 4. Syntonic block of nineteen notes.
Figure 5. Magic block of nineteen notes.
Major and minor reverse-‐mappings Depending on which of the two lines separated by the unison vector are chosen as being included in the block, the resulting selection has a bias towards major triads or towards minor triads. For example, in the syntonic example above if we include the D that is 3/2 above G we get the JI scale in row 1 of Table 1 (which has three major triads two and minor triads); if we include the D that is 3/2 below A we get the JI scale in row 2 of Table 1 (which has three minor triads two and major triads). The difference between the major and minor reverse-‐mappings is simple—the pitches of the notes will be identical with the exception of the reference tone which will differ by a comma. Ideally the user would be able to toggle between major and minor mappings, and it would probably make sense for the frequency of the reference tone to go either up or down (depending on the position of the major/minor toggle) by half a comma as the mapping slider is moved from fully tempered to fully JI.
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3. What control does this give the user? Imagine two sliders: • An Overtone Tempering slider which controls the degree of tempering applied to the spectra of every tone, so that at its leftmost position tones have fully harmonic spectra, at its rightmost position the spectra is fully tempered to the current tuning. This control is implemented in WickiSyn. •
A Fundamental Tempering slider, which controls the degree of tempering applied to the tuning of the fundamental of every tone with respect to the reverse-‐mapped JI block as derived above, so that at its leftmost position tones are tuned to JI, at its rightmost position the tones are fully tempered to the current tuning.
The absolute and relative positions of the two sliders have an impact on three perceptual effects: harmonicity, regularity, and consonance. We get: • Full harmonicity when the Overtone Tempering slider is fully to the left (overtones are harmonic) • Full regularity when the Fundamental Tempering slider is fully to the right (fundamentals are aligned with the notes of the current tuning of a rank-‐2 regular temperament) • Full consonance when the two sliders are in the same position (overtones and fundamentals all align with the current tuning’s scale steps, whether regular or JI) If we assume that maximal harmonicity, regularity, and consonance are “good” things, then it is clear that we can only ever get two goods out of the three—but we can optimise one of them and trade between the other two, as follows: 1. Set the Overtone Tempering slider to the left giving full harmonicity, and trade between consonance and regularity by adjusting the Fundamental Tempering slider 2. Set the Fundamental Tempering slider to the right giving full regularity, and trade between consonance and harmonicity by adjusting the Overtone Tempering slider 3. Lock the timbre and mapping sliders together to give full consonance, and trade between harmonicity and regularity by adjusting both sliders together. So you can pick your poison (or precise mixture of)!
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The Tone Triangle From a GUI perspective the above relationships can be nicely illustrated (or possibly controlled) using a triangular “pad”:
Figure 6. The triangle of perception
In the above example, the position of the star indicates a setting with high regularity and consonance but low harmonicity. The distance of the star relative to the harmonicity line corresponds to the position of the Overtone Tempering slider, the distance of the star from the regularity line corresponds to the position of the Fundamental Tempering slider, while the distance of the star from the consonance line corresponds to the distance apart of the two sliders. The diagram illustrates how the X_System, as originally implemented, lived entirely in the bottom right corner of this triangle. The addition of the Overtone Tempering slider in WickiSyn2 enabled us to traverse the bottom line of the triangle. The addition of a new Fundamental Tempering slider would enable the entire perceptual triangle to be explored.
The Tone Diamond Because there are a two possible reverse mappings—one to a selection of JI tones favouring major harmony and one favouring minor harmony we can use two triangles joined together to form a diamond shaped control/display pad. This diamond enables us to smoothly navigate the entire range of tonal possibilities.
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Figure 7. The Tone Diamond.
The circles in the figure above indicate notable positions: •
• •
the circles on the diamond’s points indicate positions where two percepts are optimised at the expense of the third (e.g. JI with harmonic timbres, which optimises harmonicity and consonance at the expense of regularity); the circles on the lines indicate positions where one percept is optimised and the other two percepts are equally de-‐optimised; the circles in the middle indicate a position where all three percepts are equally de-‐ optimised—in this position the system is only one third away from being fully harmonic, one third away from being fully regular, and one third away from being fully consonant.
4. Playing in JI, and exposing the linkage between regular temperament MOS scales and their JI archetype This system allows the Thummer-‐player to play in JI precisely as she would play in regular temperaments (something Allaudin Mathieu asked for), and the JI explorer has the benefit of being able to invoke a different selection of JI notes by choosing a different regular temperament. Assuming the regular temperament is a sensible one (i.e. the comma is relatively small in size), the resulting JI note selection will also be a sensible one because it will have a high density of triads clustered together. Furthermore, the notes of this JI note selection will retain the Thummer keyboard’s geometry, so changing between a regular temperament and its reverse-‐mapped JI equivalent will not require changing fingering.
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Moving the Fundamental Tempering control between JI and tempered will also allow for the user to hear the relationship between a temperament’s MOS scales and the JI archetype from which they are derived. In this way we can see and hear how each different MOS scale is a specific selection of notes taken from JI such that, with a small amount of tempering, they become organised into a simple and aesthetically pleasing pattern (i.e. two step sizes distributed with maximal evenness, and having constant structure). For example, in the syntonic temperament, the tempering out of the syntonic comma allows for all of the notes making up six tightly clustered major and minor triads to be connected with a simple seven note (diatonic) MOS scale. In a different temperament like Magic the tempering out of the comma allows for a different selection of tightly clustered JI triads to be connected with a different MOS scale.
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