Most Western musical scales, along with pentatonic scales, and others, are based on octaves repeating. In Western music, the system of octaves split into 12 logarithmic equal divisions known as “12-tone equal temperament” (hereafter abbreviated to “12-TET”) is ubiquitous, and gives rise to the set of standard scales such as “A Minor”, or “C Major” – all ultimately derived from the 2:1 and 3:2 interval ratios of the “circle of fifths“.
I am currently (2022) working with a scalic structure that does not conform in the above sense.
A Golden Ratio 833-cent scale (of which there can be be many variants) is based upon a repeat pattern of eight unequal intervals where the interval between note 0 and note 8 is 8.33090… semitones. The essential ratios are not 2:1 and 3:2 – they are 1.618034… and 0.618034… In this system the octave (12 semitones) cannot be sounded – the closest thing you might hear is approximately 11.5 semitones, which to a Western attuned ear would sound quite strange.
TABLE 1 shows the intervals of a standard-tuned western piano 12-TET contrasted against an example Golden Ratio tuning. This particular tuning was first suggested by Sevish – one of the leading proponents of micro-tonal music:
The GR tuning is created by iteratively dividing the frequency ratio of the repeat interval (8.33090… semitones) into two intervals in the proportion of the Golden Ratio or 1.618034… which is known as “Phi” or “Φ”. Thus (for example) in the first iteration, the first “split” is made at 8.33090… / 1.618034… = 5.14878…
The need for a Vocabulary
I create soundscapes using generative software and synthesizers. When generative software feeds MIDI signals to an instrument tuned to 12-TET, then there is a direct keyboard mapping between the note names in the generator, MIDI note numbers generated, and the frequencies sounded by the synthesizer. However, when using Golden Ratio (hereafter abbreviated to ‘GR’) tuning, there is no such default mapping. In addition, the chords have no default names. For example, in GR the tri-chord obtained by playing the piano keys C4 : F4 : Ab4 can in no useful way be described as:
“A badly tuned F minor with an accurate inverted fifth, but an awkwardly sharp root, and very sharpened minor third.”
This descriptor is accurate but impractical – so how can we facilitate any discussion, or develop any compositional vocabulary for GR music? One could simply take a piece of music composed for 12-TET, and replay it on a GR-tuned instrument – but what would be the point? It would just sound like normal music played on a badly tuned instrument.
But I need to program my software-based MIDI generators to be tuning-aware, and so I have striven to develop for myself a set of GR musical descriptors which together form a vocabulary that allows me to identify and conceptualize GR-based sounds, musical structures and transformations – in order to develop and progress musical ideas in a GR context.
Keyboard Key Names
I continue to label the keys on the keyboard by their traditional names – “C4”, “C#4”, “D4” and so on – however, it is essential to be aware that in GR the name of the keyboard key is no longer synonymous with the sound evoked by that name in the mind of the traditional Western musician. I must cease to think of the name “A4” as synonymous with the frequency 440 Hz – because in GR it is 454.107532… Hz, approximately half a semitone sharper.
Nevertheless, my descriptor system must retain the traditional names of the keyboard keys, because my generators understand that terminology, and my keyboard is laid out that way.
GR Interval Names
The other striking fact about GR tuning is the presence of unequal intervals. In 12-TET, the interval C4:D4 and the interval D4:E4 both express an interval of exactly 2 semitones, and the usual method of creating sounds based on scales of unequal intervals is to employ Key Signatures.
In Sevish 833 GR tuning, the interval C4:D4 is 1.96667… semitones, and the interval D4:E4 is 2.43091… semitones. In other words, the basic pattern has unequal intervals “out of the box“.
So the first step in inventing suitable descriptors is to find some reasonable names for the new intervals, so that we don’t have to keep quoting irrational decimal numbers to an arbitrary length. I propose that it is reasonable to name the basic intervals using the first three digits of their decimal expansion, thus:
“121”, “196”, “318”, “439”, “514”, “636”, “711”, “833”
So, although we know that the accurately tuned instrument will reproduce 5.14878… semitones for the C4:F4 interval, we can nevertheless, when conceptualizing GR music, refer to that interval simply as the “514” interval.
GR Interval Patterns
In 12-TET, the Key Signatures system creates recognizable sonic signatures by a process of doubling or even trebling some of the interval widths – this is achieved by omitting some of the possible 12 notes in each octave. There are very many possible approaches to omitting frequencies, but the most common method is to simply select 8 notes out the possible 12. This gives rise to the familiar “Major”, “Harmonic Minor” and the “Melodic Minor” Key Signatures.
In addition, each interval pattern can (theoretically at least) be rotated into one of seven Modes. As an example, the seven Modes of the Major Scale are constructed by taking the Major interval pattern of whole and half tones (H), and simply rotating the start position of the sequence by one place:
- ( W-W-H-W-W-W-H ) = Ionian/Major
- ( W-H-W-W-W-H-W ) = Dorian
- ( H-W-W-W-H-W-W ) = Phrygian
- ( W-W-W-H-W-W-H ) = Lydian
- ( W-W-H-W-W-H-W ) = Myxolydian
- ( W-H-W-W-H-W-W ) = Aeolian
- ( H-W-W-H-W-W-W ) = Locrian
(Historical Note: This is just analysis after the fact. Modern music did not actually develop this way. Modes came first, and then Key Signatures were introduced in order to make music more “tonal”.)
In the 12-TET system a tonic is the first degree of a scale. The tonic defines the name of the scale, and serves as a natural resolution point for all other notes in the scale.
In GR, a “tonic” serves to identify the first note of a sequence of intervals. But it does necessarily form a resolution point, because the interval pattern that surrounds it is not built from 12-TET perfect fifths. Nevertheless, the term “tonic will be useful. We have to start somewhere… literally!
In GR music, exactly what it means to talk about a root note as a “tonic” is quite a deep question. The interval patterns of GR emerge mathematically – they have not been “designed” – for example, to contain specific tri-tone dissonances that can be employed prior to returning to the tonality of the scale.
So in a GR context I use the word “tonic” very loosely – to mean a note with its associated interval patterns to either side of the keyboard that can form the basis of a composition – a particular root frequency and a set of associated chords that the music seems to keep returning to, or re-stating in some way.
The GR Intervals
The Sevish GR tuning forms a diatonic interval pattern (in semitones) like this:
1.21… 0.75… 1.21.. 1.21… 0.75… 1.21… 0.75… 1.21…
I can’t use the style W-H-W-W-H-W-H-W to represent this, because these intervals are not “wholes” or “halves” of anything, so I use W for “wide” and N for “narrow” – the starting pattern is therefore:
If I tune my keyboard such that C4 (Middle C) sounds the traditional 261.63 Hz, and designate that as my “tonic”, and thereafter tune to the GR repeating 8-interval pattern, then I have defined (in some sense at least) a GR scale.
Therefore in my GR vocabulary, I define that there exists a GR scale (lets call it “GR Major” for lack of a better terminology) plus 7 other GR Modes. Here I am adapting the naming convention from 12-TET modes, because they are constructed in exactly the same manner – i.e. By simply rotating the interval pattern of the basic scale by one position.
This naming convention has the advantage that (for example) a diatonic transposition from the GR C4 Major scale which moves notes up by 2 intervals, will transpose the composition into GR Phrygian Mode (W-W-N-W-N-W-W-N) – and I know this by remembering the sequence from 12-TET.
However, I have 8 Modes rather than 7, so I need one more name, which I will arbitrarily call the “GR Stygian” Mode.
I refer to successive intervals from the tonic by a count “1”, “2”, “3” etc. – but it is important to bear in mind that these are only interval counts. Thus the fifth interval is literally what it purports to be (i.e interval number 5 in a sequence of 8 intervals), and is not to be confused with a 12-TET “fifth” i.e. the interval from the tonic to the note that closely approximates a 3:2 interval ratio, but is actually the seventh note.
Thus, when I diatonically transpose a GR composition (moving all notes up or down a specific number of intervals), it does not change Key, it changes Mode. A melody played commencing at C4 will sound like a GR Major/Ionian Mode (W-N-W-W-N-W-N-W). If you play the same melody starting at E4, it will sound like the GR Myxolydian Mode (N-W-N-W-W-N-W-W).
Locating the Interval Patterns
I cannot memorize the interval patterns to both sides of every note on a GR tuned keyboard, so I use the method of locating the Major roots and counting upwards. Remembering where the Major roots are is quite simple. Here is the pattern:
C-1, G#0, E1, C2, G#2, E3, C4, G#4, E5, C6, G#6, E7, C8.
So the answer to “What mode am I in if I use F#2 as a tonic?” is “Well, C2 is Major, so transposing up 6 intervals puts you in GR Locrian.”
GR Key Signatures ?
We could also create very many other interval patterns in GR by the familiar process of omitting some frequencies. One could create a set of 6-note or 5-note scales for GR – but I will not go into that additional level of complexity in this exposition.
GR Mode Characteristics
I have arbitrarily defined the GR Major/Ionian Mode as the sound heard when I play melodies or chords from the interval pattern W-N-W-W-N-W-N-W.
In this mode, there is the highest likelihood of hearing sounds that embody the Golden Ratio 1.618034… or “Φ”.
Golden Ratio sounds will be heard when sounding intervals (121:196, 196:318, 318:514 and 514:833) – or, expressed as interval counts, (1:2, 2:3, 3:5, 5:8) – the Fibonacci series numbers.
As can be seen in TABLE 3, all of these intervals can be heard in the GR Major, where the (4:7) also sounds the Golden Ratio.
In other GR Modes, some of the characteristic intervals cannot be played – the highlighted intervals in TABLE 3 are those that arise out of the pattern rotation, and introduce different sonic signatures.
If we consider the GR Myxolydian, you will not hear the intervals 121, 439 or 636 (which is arguably no big deal) but the 318 is also missing, so something of the “defining” Fibonacci based 318, 514, 833 Golden Ratio sound will be absent.
GR Stygian is curious indeed, because the 5th interval (that tantalizingly “almost a 12-TET Fourth”) is missing. The (2:3), the (3:5), the (4:7) will also not sound as expected from the basic scale, giving the Stygian a distinctly different feel.
A composition in GR Lydian sounds very much like the GR Major, but gives the opportunity to forego the (4:7) chord and sound the Golden Ratio more unusually as a (4:6).
GR Chords & Chord Naming
I arbitarily define a GR Chord as one which is formed by selecting a root note, adding a second note (call the interval between them “N” cents), and then adding on top a third note whose interval is
N x Φ cents
which is to say,
N x 1.618034…
The “examplar” GR chord of the system is the sound heard when playing the keyboard keys:
C4 : F4 : G#4
The intervals in semitones between its components are:
- lower interval C4 : F4 = 5.14878…
- upper interval F4 : G#4 = 3.18212…
- outer interval C4 : G#4 = 8.33090…
and it produces the frequencies:
- C4 = 261.63 Hz.
- F4 = 352.25 Hz.
- G#4 = 423.33 Hz.
and sounds like this:
Compare it to the 12-TET F Minor inversion (C4 : F4 : Ab4) with semitone intervals:
- lower interval C4 : F4 = 5.0
- upper interval F4 : G#4 = 3.0
- outer interval C4 : G#4 = 8.0
that produces the frequencies:
- C4 = 261.630 Hz.
- F4 = 349.23 Hz.
- G#4 = 415.30 Hz.
and sounds like this:
If your ears are not accustomed to listening for minor differences in pitch, then play both of these recordings at the same time. You will hear the additional peculiar sound of beats at 3, 5, and 8 cycles per second – more Fibonacci numbers! – (The beats are not real sound, your brain just conjures them out of the differences in frequency).
The GR interval arrangement generates the (to me!) beautiful, complex and infinite set of Fibonacci partials and overtones that define the Golden Ratio sound. The effect becomes even more pronounced if you employ instrumentation with rich timbres.
I have designed for myself a terminology for specifying such GR tri-chords:
- Name the root using its keyboard key name.
- Specify the other two notes involved in the triad using a count of how many intervals exist in between that note and the root (regardless of the actual interval sizes).
- Append the associated GR Mode name as an abbreviated comment. This is just for reference – it helps me to remember what interval pattern lies either side of the root note.
Specified in the new terminology, C4 : F4 : G#4 becomes:
[C4 +5 +8)]//Maj
As a naming system, it is compact and functional. It also has a major advantage for me, in that having written it down in exactly this format, I can then “cut & paste” the chord directly into my favorite sequencer, “Nodal” – a tool that understands this written form.
The chord naming system tells me everything I need to know:
- That its root is C4.
- That it is a Major chord, so pitches relative to the root note conform to the interval pattern W-N-W-W-N-W-N-W.
- That it is played on a keyboard as C4 + the note 5 piano keys higher + the note 8 piano keys higher.
- That since it is a Major, and a 5:8, it will sound the Golden Ratio.
[E3 +8 +13]//Maj also sounds the Golden Ratio – but will sound more relaxed because the intervals are wider:
[E3 +3 +5]//Maj will also sound the Golden Ratio – but will sound more tense , because the intervals are narrower:
[E4 +3 +5]//Myx will not sound the Golden Ratio, because it 3rd interval is 271, not 318. GR Myxolydian mode has a narrower 3rd interval.
That is not to say that E4 Myx (3:5) is unusable – far from it – it is an interesting chord. Consider this progression:
[C4 +4 +7]//Maj > [E4 +3 +5]//Myx > [E4 +5 +8]//Myx
It employs the Myxolydian (3:5) to transition from a chord in Major to another chord in the target mode. It is a gentler transition then moving more dramatically between the two chords:
[C4 +4 +7]//Maj > [E4 +5 +8]//Myx
In 12-TET, common methods of modulating between different Key Signatures include employing a chord that exists in both Key Signatures, or sometimes to move step-wise through a sequence of chords, each of which shares at least two of its notes with the preceding chord. Thus in GR, we become interested in looking for chords that are related to the GR Chords by virtue of having at least 2 notes in common with the GR chord under consideration.
Mirrored GR Chords
I also (again quite arbitrarily) define a mirrored chord to be a GR chord where the outer notes are retained, but the middle note has been moved so as to invert the interval ratios.
In 12-TET, a Major chord is converted to a Minor by simply flipping the third downwards, so that a (4:7) semitone interval ratio becomes a (3:7) ratio. However, the term “Minor” could be confusing, so I adopt the term “mirrored” – because this what the interval ratio looks like when viewed in an imaginary mirror.
Mirroring a (5:8) takes the ratio from (5:8) to (3:8). Thus we have a series of GR mirrored chords such as :
- [C4 +3 +8]//Maj
- [C#4 +2 +5]//Dor
- [D#4 +3 +8]//Lyd
So, mirrored GR chords sound the inverse Golden Ratio, which is the second “defining” sound of GR music.
Here is a 4-chord modulation from Lydian to Dorian via a mirrored chord. You can tell that it ends in Dorian because the Dorian (4:7) is characteristically not a Golden Ratio – at 1.81 it is actually closer to a 5/4 ratio.
[D#4 +5 +8]//Lyd > [C4 +5 +8]//Maj > [C#4 +3 +8]//Dor > [C#4 +4 +7]//Dor
There will also be an equivalent of the “inversion” which is popular in 12-TET – namely, retaining 2 of the notes of the triad, but transposing the other note by one octave. But there are no octaves in GR – so instead I (again, arbitrarily) define inversion in GR to mean moving an outer note to a position which preserves the original interval ratios.
So our exemplar chord [C4 +5 +8]//Maj has inversions:
[E3 +8 +13]//Maj
[F4 +3 +5]//Aeo
So in GR, inversion retains the Golden Ratio sound, but shifts it up or down in pitch, widening or narrowing the intervals appropriately. I refer to the narrowing of intervals and higher pitch as an “Upper Inversion” and the widening of interval and lowering of pitch as a “Lower Inversion”.
Thus, GR inversion is not simply a “variant” of the original chord, it is a powerful modulation mechanism as well.
The GR scale is created by iteratively diving the “833” interval. So it makes sense that some GR chords can be further divided – employing another “iteration” of the Golden Ratio.
The exemplar chord [C4 +5 +8]//Maj can be divided in such a way that we retain 2 of the notes whilst interpolating a new note in between. I call these related but narrower triads the “Internal” chords. The exemplar chord [C4 +5 +8]//Maj offers potential internal chords:
[C4 +3 +5]//Maj and its mirror, [C4 +2 +5]//Maj
[F4 +2 +3]//Aeo and its mirror, [F4 +1 +3]//Aeo
These latter narrow intervals are quite strident.
Equidistant and Equal Interval Chords
Yet another variation of the “retain 2 of the notes” class of transforming triads is to depart from the GR sound by widening or narrowing one of three intervals that are present in the original triad.
There are very many ways to do this, but a special sub-class of such transformations will be those where the transformation makes the interval count equal in size – the “equidistant” related chords. Note that depending on which Mode you are in, making the interval count equal may or may not result in equal interval sizes – so there exists a further subset of the Equidistant chords that will also be Equal Interval chords.
So again using the exemplar chord:
[C4 +5 +8]//Maj can be transformed it into one of six Equidistant Chords:
Balancing the outer interval gives either:
[E3 +8 +16]//Maj or [C4 +8 +16]//Maj
Balancing the lower interval gives either:
[G3 +5 +10]//Lyd or [C4 +5 +10]//Maj
Balancing the upper interval gives either:
[D4 +3 +6]//Phr or [F4 +3 +6]//Aeo
All of these are Equidistant chords, and are also Equal Interval chords, because in all Modes except GR Dorian and GR Myxolydian, a (3:6) is Equal Interval.
Here is a chord progression employing an Equal Interval to modulate from GR Lydian to a mirrored GR Major chord:
[G3 +5 +8]//Lyd > [G3 +5 +10]//Lyd > [C4 +5 +8]//Maj > [C4 +3 +8]//Maj
Of course there a very many other chords playable, but I personally don’t care to classify or name them any further.
Expanded GR Chords
The naming system automatically caters for chords with any number of constituent tones, for example, a Myxolydian archetype:
[G#3 +3 +6 +9 +11 ]//Myx
or the characteristically Stygian sound of a (2:5:7) chord:
[B3 +2 +5 +7]//Sty
So, by virtue of this vocabulary, I have a descriptor system that allows me to:
- Conceptualize any GR chord.
- Name it and write it down.
- Know which interval patterns lie to either side of its root.
- Imagine what it might sound like.
- Write, or “cut-and-paste” it into my sequencer in order to generate the appropriate MIDI messages to sound that chord on a GR tuned synthesizer.
- Know which physical keyboard keys will reproduce it.
And moreover, I have numerous options for:
- Generating melodies and chords within a mode.
- Transcribing compositions.
- Modulating to a different mode, and back if required.
GR Music EXAMPLE
Some people are uncomfortable with GR music, others say that the lack of tonal references creates a wistful or other-worldly atmosphere. All the tracks of my 2022 soundscape “Lovecraft Ambience III” use the “Sevish” Golden Ratio tuning and scale described above.