Types of Tunings You Can Make |
Equal Divisions of Ratios |
Srutis
In this application, a
sruti is an equal division of another interval,
sometimes called a
chromatic step (John Pierce uses that term).
In Indian music theory, the octave is divided into 22 srutis and scales
are formed by taking a subset of these 22 notes. Indian srutis are not
actually equal-sized, or even exactly specified although they are treated
as if they are equally spaced. In concept however, the word sruti comes
closest to explaining the main units used in LMSO, which are equal-sized divisions
of a ratio that are subsetted to form scales. In other words, a sruti is
a unit of subdivision.
| “Stretched octave” tunings |
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A stretched octave tuning is used when tuning pianos.
The stretched partials are used to assist in tuning the stretch,
but it is not this in-tuneness that makes stretched tuning preferred by
listeners.
Skeptical? Not what you’ve been told by the psychoacoustechnicians?
To prove this to yourself, compare stretched tuning to just-octave tuning using a
non-stretched timbre, like a sawtooth or square wave. Since the timbre doesn’t
have stretched partials, the stretched tuning will make the partials
less “in tune”, not more (as it does with a piano’s naturally stretched
partials). Yet hear how the stretched tuning still sounds better than the flat tuning.
So timbres without stretched harmonics sound just as good when stretched!
This shows that the proof is in listening, not in the theorizing and posturing
while inhabiting lofty clouds.
So why do stretched octaves sound better? Partials beat together
at a pleasant rate, causing shimmer. Shimmer is more complex than vibrato since it has more components.
The more components (partials), the fancier (more complex) the shimmer.
It is the solvent of shimmer, which in a small way breaks up the stagnant clods
that justly tuned octaves precipitate in your harmonic solution.
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Octave Tunings
In the Oven’s
Recipe area, the “Scale Pattern” edit field expects a comma-separated
list of #s of srutis.
Thus, to specify twelve tone equal temperament (12tET) — the tuning most
commonly used in the modern western world, one would set the Repeat Ratio
to
2:1 and type
1,1,1,1,1,1,1,1,1,1,1,1 as the scale
pattern. Or, to specify a diatonic major 12tET scale, one could type
2,2,1,2,2,2,1; resulting in:
Similarly, diatonic 12tET minor has a Scale Pattern of: 2,1,2,2,1,2,2.
For a slightly stretched
octave tuning, simply specify the Repeat Ratio to be in the range 2.002 to
2.017 or so. You will find that by stretching the octave,
the same just approximations will be captured, but the tuning will sound
more vibrant now that those dull octaves have some shimmer to liven them up.
12tET with 1213 cent (stretched) octave
Another way to specify 12tET is to set the Repeat Ratio to 2^(1/12)
and set the scale pattern to 1.
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(Yes, you can type mathematical
expressions into any edit field that expects a number. You can also use
the up and down arrow keys on numbers to increment and decrement them —
by a factor of 10 when the shift key is pressed. Such attention to detail
combined with genuinely useful features are hallmarks of Red Barn Goat Farm’s products.)
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But subsetting 12tET is not what the Li’l Miss is about. She’s up and about
showing that the octave is meaningless and that just intervals are not
necessary or special. Octave preference is learned; not an intrinsic part
of our wetware (brains). Awareness of the existance of just intervals is
useful to help create beating frequencies between the overtones of notes
that are played together. The just intervals are particularly relevant
when playing waveguide instruments such as wind instruments and strings
whose overtone series are comprised of roughly integral multiples
of the fundamental frequency.
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Just intervals are frequency ratios that can be specified with relatively
simple fractions — like a fifth when tuned exactly to 3/2, a fourth tuned
exactly to 4/3, or an octave tuned to 2/1. Some composers and theorists
believe that the use of just intervals in composition and performance
allows a higher level of music, or induces special mental or emotional
states. I find perfectly tuned simple intervals to sound rather static
and annoying on electronic instruments that are capable of maintaining
perfectly phase-locked harmonies.
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Some fascinating tunings that can be easily created within the sruti framework are now
discussed...
88 cent Equal Temperament (cET)
An 8-way-divided fifth gives a sruti 87.745 cents in size — very close
to 88 cET, a popular nonoctave tuning discovered by Gary Morrison. 88cET
is popular and useful because it contains near-just hits for 10:9, 7:6,
11:9, 7:4, 9:7, 10:7, 3:2, 5:3, 15:7, 9:4, 5:2, etc.
Gary uses exactly 88.000cET, but you can also try a harmonic seventh interval
divided equally 11 ways — the 11th root of 7/4, or 88.075cET.
88 cent Equal Temperament, Javanese mode
This subset of the 88cET chromatic scale distributes the 11 divisions of 7/4
as [3,3,2,3] (absolute: [0,3,6,8,11]). This gives the following scale which hits close to a number of interesting
intervals, with enough detuning to shimmer them (shimmer is critical in Indonesian tuning, where
purely tuned intervals are eshewed as being dead sounding):
(7:4)0/11 = 1/1
(7:4)3/11 = 7/6 - 2.645658 cents
(7:4)6/11 = 9/7 + 5.291317 cents
(7:4)8/11 = 3/2 + 2.645658 cents
(7:4)11/11 = 7/4
(7:4)14/11 = 2/1 + 33.051154 cents
(7:4)17/11 = 9/4 + 5.291317 cents
(7:4)19/11 = 13/5 + 19.212618 cents
(7:4)22/11 = 3/1 + 35.696812 cents
(7:4)25/11 = 18/5 - 15.719228 cents
(7:4)28/11 = 4/1 + 66.102307 cents
This scale works well with Chocolate Frosting (map it to the black keys).
The Bohlen-Pierce Scale — a Twelfth Tuning
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| chromatic BP scale
| “diatonic” BP scale
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Pierce split the twelfth (3:1) 13 ways to create a tuning based on a 146.3
cent equally tempered “chromatic scale”. It features near-just hits for
9:7, 7:5, 5:3, 9:5, 15:7, 7:3, 11:4 and a just 3:1 (although Pierce didn’t
notice the 11:4 and 15:7, or even the near-12:11 [12:11 = 150.6 cents]
base of the scale.) In the terminology used here, 13th-root-of-three is
a scale derived from a 146.3 cent sruti base. It could also be called “146.3
cent equal temperament”. Pierce subsetted the chromatic scale to feature
near-just intervals by specifying a repeat ratio of 3:1 and scale pattern
of 3,1,2,1,3,3. I would recommend using 3,1,2,1,2,1,3
instead in order to include the 7:5 down from tonic (15:7 up from tonic).
13th Root of Minor Just Tritone (7/5)
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Adjectives
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O p e n i n g
T h o u g h t f u l
R e f l e c t i v e
H e a l i n g
I n v i g o r a t i n g
E x c i t i n g
B r i g h t
T h r i l l i n g
B r e a t h l e s s
F r e e
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Good Timbres
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Werkmeister Piano
Brass Ensemble
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Associations
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Spleen Meridian
Crown Chakra
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The default scale is called “Just Tritone-13” or “JTT/13”. It is based
on the 13th root of the interval 7:5, the just tritone (also called the
septimal tritone). The 4,1,1,1,4,2 subdivision pattern permits
the tuning to be mapped straight across the keyboard in a very useful and
easy to play way. This tuning is very nice because of the following properties:
{0,4,5,6,7,11,13} => {4,1,1,1,4,2}
(7:5)0/13 = 1:1
(7:5)4/13 = 10:9 - 3.2 cents
(7:5)5/13 = 8:7 - 7.1 cents
(7:5)6/13 = 7:6 + 2.0 cents
(7:5)7/13 = 6:5 - 2.0 cents
(7:5)11/13 = 4:3 - 5.2 cents
(7:5)13/13 = 7:5
(7:5)15/13 = 10:7 + 6.6 cents
(7:5)17/13 = 3:2 + 5.4 cents
(7:5)21/13 = 5:3 - 10.6 cents
(7:5)23/13 = 7:4 - 11.8 cents
(7:5)25/13 = 11:6 - 9.2 cents
(7:5)26/13 = 13:7 - 10.1 cents
(7:5)29/13 = 2:1 + 6.6 cents
(7:5)34/13 = 9:4 + 10.8 cents
(7:5)35/13 = 7:3 - 10.6 cents
(7:5)38/13 = 5:2 - 5.2 cents
(7:5)40/13 = 12:7 - 12.4 cents
(7:5)41/13 = 8:3 - 7.9 cents
(7:5)49/13 = 13:4 - 1.7 cents
Recipe
Repeat Ratio: 7:5
(1 Sruti = 44.81 cents)
Scale Pattern: [4, 1, 1, 1, 4, 2]
So we have a a 2 cent sharp subminor 3rd (7:6), a 2 cent flat minor
3rd (6:5), 5 cent flat 4th (4:3), a just tritone (7:5), a 5-cent sharp
fifth (3:2), and a very wide variety of other intriguing near-just intervals.
I consider this scale to be far more consonant than 12tET and far more
useful than just tunings. It has a jazzy feel, probably in part because of
the jazzy subminor third and the cool, calm, laid-back pure tritone
as the base.
15th Root of Minor Just Tritone (7/5)
Hauntingly beautiful. Tremendous possibilities. Highly recommended.
This is another 7:5 based scale. Some studies have shown that people find the
interval of 7:5 to be one of the most consonant and pleasing intervals. The mode
I like to use:
{0,4,6,7,8,9,13,15} => {4,2,1,1,1,4,2}
...has many very accurately tuned intervals:
(7:5)0/15 = 1:1
(7:5)4/15 = 12:11 + 4.7 cents
(7:5)6/15 = 8:7 + 1.8 cents
(7:5)7/15 = 7:6 + 5.0 cents
(7:5)8/15 = 6:5 - 5.0 cents
(7:5)9/15 = 11:9 + 2.1 cents
(7:5)13/15 = 4:3 - 5.2 cents
(7:5)15/15 = 7:5
(7:5)21/15 = 8:5 + 1.8 cents
(7:5)22/15 = 18:11 + 1.8 cents
(7:5)23/15 = 5:3 + 8.8 cents
(7:5)24/15 = 12:7 - 1.1 cents
(7:5)36/15 = 9:4 - 5.9 cents
(7:5)39/15 = 12:5 - 1.1 cents
(7:5)49/15 = 3:1 + 0.9 cents
When mapped to a chromatic keyboard, the fifth-hand-spacing
always gives 7:5 and the octave-hand-spacing sometimes gives 12:7.
The following subset is also of interest:
(7:5)0/15 = 12:12
(7:5)4/15 = 12:11 + 4.7 cents
(7:5)8/15 = 12:10 - 5.0 cents
(7:5)13/15 = 12:9 - 5.2 cents
(7:5)24/15 = 12:7 - 1.1 cents
(7:5)39/15 = 12:5 - 1.1 cents
(7:5)49/15 = 12:4 + 0.9 cents
...since it contains an excellent approximation of much of the 12-utonal
scale (the undertone series with 12 as numerator), but conveniently leaves
out the dull intervals of 12:8 & 12:6 — the fifth and octave.
15th root of 7:5 does have similarities to 13th root of 7:5, but it is different.
It is more symmetrical. It is not as earthy and grounded. It is more forceful,
but in a smooth and elegant way. You are being coerced, but it is with your
full consent. It can grab you and take you along on a bumpy wooden roller-coaster
through a melange of spine-jarring dips and turns of disjointed emotions and psychic
states with each change in harmony. Most succinctly: Effective affectiveness.
5th root of Subminor Third (7/6)
14th root of Supermajor Sixth (12/7)
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Adjectives
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S h i m m e r i n g
C r y s t a l l i n e
P e a c e f u l
H o l y
R e s t i v e
Spherical crystal enclosure
M y s t i c a l
L e v i t a t i n g
In the Groove of Intonation
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Good Timbres
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Full Tines
Voices
Tuba Violin
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Also known as 66.652cET and also as “Celestial Interference”. We set the Scale Pattern to
3,1,2,3,2,1,1,1 to create a subset
of this interesting tuning which gives a perfect supermajor sixth (12:7) as its Repeat Ratio
and a nearly-just subminor third (7:6) that is only about a quarter cent flat.
This scale has little dissonance and does have some just octaves here and there.
Because of the interesting mathematical novelty whereby (12/7)3/14 is very close
in value to (2/1)2/12, we have some intervals that are quite close to 12tET whole tones.
Though this tuning is far removed in mood from 12tET, it can occasionally generate in passing
some reminescence of 12tET.
Because the chromatic scale is nearly identical to sixth-tone tuning, some might say this tuning
is identical to 36tET. I disagree since the octave is not the repeat interval — 12/7 is.
Since 12/7 is the dominant interval, naming this “14th root of Supermajor Sixth” tuning is
far more appropriate.
9th root of Subminor Tenth (7/3)
A nice tuning because it approximates so many 11-limit intervals but misses the octave as far as
possible. Also, the complex intervals are less ‘out-of-ratio’ than the simple ones (which can
tolerate a wider detuning while maintaining their class distinction).
I don’t find that this scale works for solo lines played in a void, but it works well
melodically when accompanied by itself, or by other scales, or by transposed copies of itself. The
mood is a little South-East Asian.
An interesting novelty in this scale is that it contains 1st inversion major triads and
2nd inversion minor triads that are ringers for Western common practice tuning, even
though this scale is totally alien to that system. The chords are not particularly usefully
arranged, so although western-sounding harmonic progressions are possible, interesting ones are
elusive. It may be amusing to fool-your-friends with this tuning at parties since few people will
hear anything at all unusual about the tuning in which you are playing perfectly normal
sounding major and minor chords until they try to play it themselves — then the fun begins, ho ho!
1st inversion major triad in 9th root of 7/3:
0 1/1
2 6/5 + 10.330025 cents
5 8/5 + 1.241995 cents
2nd inversion minor triad in 9th root of 7/3:
0 1/1
3 4/3 - 9.088031 cents
5 8/5 + 1.241995 cents
Of course, there are several other conventional sounding chords that can be found
in this tuning if that’s your thing.
8th root of 11/7
This tuning works very well with piano and double-struck guitar and koto timbres.
With such timbres, it features beat frequencies that are reminescent of vibrato
traditionally used upon Japanese kotos.
11:7, at 782.492036 cents, is a bit flat of the equally tempered minor sixth at 800.0 cents and
quite a distance from the just minor 6th, 8:5, which appears at 813.686286 cents.
Since it is a ratio of small integers, it sounds perfectly “in tune”.
The 11 against 7 pattern is more vibrant, complex, and mature than the 8 against 5:
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| 11 against 7 |
8 against 5 |
12tET’s 800.0 cents sixth is about 14 cents flat or 18 cents sharp from just, depending on which interval you are comparing it to.
In any case, the minor sixth is definitely out of tune in 12tET.
Furthermore, its perceptual definition is ambiguous since it is almost equidistant between two different simple ratios.
This 11:7 based tuning is a substitute for 12 tone temperaments, but with more richness, texture, elegance and maturity.
Its intervals map to a piano keyboard in such a way that you can play many familiar chords.
But each chord sounds a little different — though no less “in tune” — than 12tET, and sometimes more in tune, or at least more interesting.
In the following table, note that 11:7d8 has many intervals that are only a few cents off from just intervals — close enough to create a delightful and invigorating shimmer.
Next, interval for interval, compare this tuning against 12tET.
Notice in particular that the fifth, octave and minor ninth are far from just, but that the other intervals
are pretty close to just. On the way to the minor tenth, 12tET has nine intervals that display
significant tuning error (greater than 10 cents), whereas 8th root of 11:7 has only four.
In fact, this tuning can be said to be considerably more consonant than 12tET, especially for
chromatic (read interesting) musics.
8th root of 11:7
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12th root of 2:1
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00 1/1
01 14/13 - 30.486740 cents
02 9/8 - 8.286993 cents
03 13/11 + 4.224794 cents
04 5/4 + 4.932304 cents
05 4/3 - 8.987477 cents
06 7/5 + 4.356834 cents
07 3/2 - 17.274469 cents
08 11/7
09 5/3 - 4.055173 cents
10 7/4 + 9.289138 cents
11 13/7 + 4.224794 cents
12 2/1 - 26.261946 cents
13 13/6 - 67.023103 cents
14 11/5 + 4.356834 cents
15 7/3 + 0.301662 cents
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00 1/1
01 14/13 - 28.298245 cents
02 9/8 - 3.910002 cents
03 13/11 + 10.790281 cents
04 5/4 + 13.686286 cents
05 4/3 + 1.955001 cents
06 10/7 - 17.487807 cents
07 3/2 - 1.955001 cents
08 8/5 - 13.686286 cents
09 5/3 + 15.641287 cents
10 16/9 + 3.910002 cents
11 13/7 + 28.298245 cents
12 2/1
13 13/6 - 38.572661 cents
14 9/4 - 3.910002 cents
15 12/5 - 15.641287 cents
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Harmonic Talk
Because the minor sixth is in tune, many 1st inversion type chords are possible.
Because the just tritone of 7:5 is almost in tune, many jazz chords are quite smooth and palatable.
Other notable intervals are the harmonic 7th (7:4), the major third (5:4), and the subminor
tenth (7:3) — all much more in tune than is possible in 12tET.
Also note that instead of a minor third of 6:5, we have one at 13:11.
Since 13:11 is a perceivable small-prime ratio and is closer to 300 cents than 6:5, people listening to
a 12tET minor third at 300 cents are more likely to hear a mistuned 13:11 than they are 6:5 anyway
(though many people just perceive a non-consonance in the 300 cents interval,
in contrast to the european public of 350 years ago who heard 300 cents as a strong and unacceptable
dissonance).
The 17 cent flat-from-just fifth is also interesting. Theorists might try to tell you such a
fifth is unusable. Try playing it in the bass register of a piano.
It sounds really great — better than a just fifth.
Creative visualization time.
Visualize the theorists living inside your ear, telling you what you are hearing.
Take those theorists and technicians, put them in a bubble and blow that bubble away.
Watch it float away... happy bubble!
Now that you’ve got your ears cleaned, you’ll find that you can hear more clearly.
This also illustrates a general principle. It is easier to hear 5 cents detuning
on a simple intervals like 2:1 and 3:2 than it is with a more complex interval like 6:5.
The beating is just more obvious. Theorists will tell you that this means simple intervals
must be tuned closer to pure just ratios than complex intervals. But they are jumping to
conclusions. Down boy! No jumping! I say the opposite is true. The mistake people make
is confusing the concept of easy-to-detect with interesting or pleasant
to listen to. There is no relationship between these two different things.
In fact, the complex intervals are more interesting when
closely in tune because it doesn’t take much detuning to make them unrecognizable.
An unrecognized interval just drops in to the harmonic background and doesn’t have much to say.
A recognized (captured) complex interval adds emotional power and subtlety to harmony.
Simple intervals can be detuned a lot before they become unrecognizable.
A detuned simple-interval shimmers and shines, adding thickness and motion.
A just simple-interval sits there in a chord like a concrete block, motionless, unnoticed, almost undetectable.
Listeners and tuners show a decided preference for octaves that are 12 cents sharp. The 1212 cents octave sounds
more smooth, more consonant, more alive, and less dead to just about everyone.
Modal Mix
A subset from a set of scale degrees is a mode of that scale.
One useful mode in 8th root of 11:7 is:
2,1,1,1,1,2
...as a
Vanilla Flavoured Frosting (mapped to the white keys).
Using modes is especially helpful to blast out any “familiar” fingering patterns you’ve gotten used to.
Also, a mode will really “bring out” the base of the scale because it imposes a repeating intervallic pattern
that your brain can’t help not just notice, but grab on to.
The chromatic 11:7 scale can sound a lot like 12tET sometimes
even though the Repeat Ratio is supposed to be 11:7.
The problem is that — with every note the same distance apart — it’s easy to rely upon the black
and white keyboard pattern to guide you instead.
But there is no mistaking this mode for 12tET. It is clearly based on 11:7.
This mode is a blast to play in. The 7-ness (septivity) and 11-ness featured in
many of its intervals is well-nigh apparent. It’s easy to stumble upon orgiastic,
kundalini-inspiring riffs.
Seven In Five
This is a just tuning based on the pattern 9/8, 9/7, 4/3, 3/2. The
scale repeats at the 3/2 so it continues 3/2, 3/2 * 9/8, 3/2 * 9/7, 3/2 * 4/3, 3/2 * 3/2 etc.
Being based on fifths and having pure fourths gives it an early music feel —
similar in feel to Pythagorean tuning since all the fifths are tuned pure just. But
only some of the octaves are tuned pure.
If played on a black & white keyboard, this works well mapped to the white
keys (Vanilla Flavoured Frosting).
Nuevo Renaissance
Yep, this is just a stretched version of the just
seven in five tuning, above. The fifth is stretched to be
8 cents sharp and the other intervals are stretched proportionately. There are
no purely just intervals here. Intervals are tuned to be a little sharp in order to give
a little edge to them.
Works well with reed instruments such as clarinet and oboe, or a recorder.
Septimal Heaven
This scale is: 1/1,7/6,7/6*7/6,7/5,7/6*7/5,7/4,7/6*7/6*7/5,7/3.
Obviously it uses the subminor third (7:6) extensively.
Entrancing and meditative when used with the right timbre, such as certain sorts of bells.
Can sound middle-eastern, but not quite. As if it is from a completely alien culture.
Can be unsettling with other timbres — very sensitive to the instrument it is played on.
If played on a black & white keyboard, this works well mapped to the white
keys (vanilla frosting).
Empirical & Cents-based Tunings |
Before you start inventing tuning systems of your own, you might experiment
with traditional western and ethnic tunings. These tunings can be found
in various
publications. (A bit of advice: you should take the
accuracy of many published ethnic tunings with a grain of rock salt.)
Sometimes you have to do a little conversion before you can use
what you have found.
Many scales published in books, journals and magazines are listed with interval
sizes given in cents. LMSO allows you to specify cents as the difference from
the previous scale degree, or as an absolute distance from the scale’s tonic/anchor.
Converting published offset scales
If the scale you have is given as an offset from twelve-tone equal temperament
(12tET), you will have to convert it. Perhaps the easiest way to do this is to
make a table of absolute 12tET values and subtract the given amount, then enter
this scale with the Scale Pattern type (Menu:Scale Pattern->Format Type) set to
Cents Absolute.
Example: Three Well-Temperaments from Owen Jorgensen’s Tuning (1991).
| Equal-beating Rousseau (1768) | Handel | Valloti (1781) |
| 12tET Note | cents absolute | Offset from Equal | Absolute | Offset from Equal | Absolute | Offset from Equal | Absolute |
| A | 1200 | +0.00000 | 1200.000 | +0.0000 | 1200.000 | +0.000 | 1200.000 |
| G# | 1100 | +2.60396 | 1102.604 | -0.5474 | 1099.4526 | +1.955 | 1101.955 |
| G | 1000 | +2.15761 | 1002.158 | +1.7204 | 1001.7204 | +3.910 | 1003.910 |
| F# | 900 | -0.23471 | 899.765 | -2.5806 | 897.4194 | -1.955 | 898.045 |
| F | 800 | +4.56346 | 804.563 | +3.4408 | 803.4408x | +7.820 | 807.820 |
| E | 700 | -0.36207 | 699.638 | +0.0782 | 700.0782x | -1.955 | 698.045 |
| Eb | 600 | +3.74226 | 603.742 | +1.4076 | 601.4076 | +3.910 | 603.910 |
| D | 500 | +1.51481 | 501.515 | -0.0782 | 499.9218x | +1.955 | 501.955 |
| C# | 400 | +1.26123 | 401.261 | -2.5024 | 397.4976 | +0.000 | 400.000 |
| C | 300 | +4.09201 | 304.092 | +4.4574 | 304.4574x | +5.865 | 305.865 |
| B | 200 | -1.50134 | 198.499 | -0.7820 | 199.2180x | -3.910 | 196.090 |
| Bb | 100 | +4.06270 | 104.063 | +2.4242 | 102.4242 | +5.865 | 105.865 |
| A | 0 | +0.00000 | 0.000 | +0.0000 | 0.000 | +0.000 | 0.000 |
0,100,200,300,400,500,600,700,800,900,1000,1100
p 155: Characters of the Keys
Cents
1/4 comma meantone
stretched 1/4 comma meantone
Just Tunings vs. Beats
Among these scales, you may notice many intervals which are close to
just intervals (small integer ratios), but a few cents off.
In my compositional view, near-just intervals are even more exciting than
perfectly tuned just intervals because of all you can do with the beats. Only when the
beats between a pair of partials is in the ‘irritation range’, does dissonance
occur. Slower beat frequencies produce different physiological, psychological,
emotional effects that are not found in purely just tunings.
The exact just interval is not what causes excitement; it is intervals
in the boundary region between consonance and dissonance. It is this
shimmering area that creates emotional impact. The feel, the intensity,
the spiritual effect, the altered states of consciousness.
2:1 is a simple easy candy interval. It is not profound. Spiritual
development becomes blocked when listening is locked to scales based in 2:1.