Is a scale cyclic or not

Hi,

Though the background is mathematical I have a quastion of more
philosophical nature that should be answered by those people that do not
model scales mathematically.

one of our students tries to find a mathematical axiomatic description
of scales, that is compatible with the algebraic theory of tone systems
as defined by Rudolf Wille. We disagree on the level where the notion of
a scale should be anchored.

On the one hand we have tones which often have a property that is called
pitch.
On the other hand we have chromas which model exactly those properties
that belong to certain classes of tones (e.g. pitch).

Both views have different aspects that are important for music. A tone
system allows to handle music as it is performed or received, while many
claims suggest that music theory mainly deals with chromas or pitch
classes. This claim often gets violated without further notice, e. G.
when octaves or nones pop up in classical western music theory in tone
systems with 12 tones per octave.

The most imortant aspect is that they give rise to tone systems, but
some aspects are not so clear to me:

1. Does a scale uniquely distinguish between octave and prime? Most
representations of scales I have seen, seem to do this
2. Why are scales often described as linearly ordered sets? Is it that
the notion of a cyclic order is not well-known to the corresponding
theorists, or does it have a deeper philosophical core.
3. Are scales anchored on a special position in the tone system as the
ancient Greek scales or medieval scales or not? Note that the tone
system itself does not necessarily have to be anchored on a physical
frequency. For example the discussion about tunings and different
concert pitches shows that the tone system of a score may be anchored at
different frequencies.

Finally,

Are scales to be understood as a set of tones or as a sets of chromas?

My intuition is that a scale is a prototypic example of a certain set of
tones, which has additional structure (order relation, interval group,
absolute intervals and so on). The student's view is that a scale is the
set of chromas of the tone system equipped with additional structure
(order relation, chroma interval group, absolute chroma intervals and so
on).

When we consider a scale a s a set of tones, it can be also considered
as rule of the construction of a tone system, while the chroma system
view suggests, that every single tone of the tone system is already
somehow (blurred and simplified) contained in the scale.

While in many applications both views have the same consequences, the
mathematical models differ significantly and have fundamentally
different properties. The question is which formalisation is better or
is used and why does the other model not provide the same information.

In order to support my point of view:
When we want to switch from the chroma view to the tone view (e.g.
turning an abstract music theoretical idea into a composition) we need a
prototypic system that allows us to generate and express the missing
information that is not available to chromas, but is important for
tones. I think scales can play this role without losing their special
identity. At least it would be very nice if the music theory already
knows a structure that can fill this gap.

Yours,

Tobias

Have a look at this page and follow the links:

http://www.lucytune.com/scales/

> On 10 Jun 2015, at 22:17, Tobias Schlemmer keinstein_junior@...t [TUNING] <TUNING@yahoogroups.com> wrote:
>
> Hi,
>
> Though the background is mathematical I have a quastion of more
> philosophical nature that should be answered by those people that do not
> model scales mathematically.
>
> one of our students tries to find a mathematical axiomatic description
> of scales, that is compatible with the algebraic theory of tone systems
> as defined by Rudolf Wille. We disagree on the level where the notion of
> a scale should be anchored.
>
> On the one hand we have tones which often have a property that is called
> pitch.
> On the other hand we have chromas which model exactly those properties
> that belong to certain classes of tones (e.g. pitch).
>
> Both views have different aspects that are important for music. A tone
> system allows to handle music as it is performed or received, while many
> claims suggest that music theory mainly deals with chromas or pitch
> classes. This claim often gets violated without further notice, e. G.
> when octaves or nones pop up in classical western music theory in tone
> systems with 12 tones per octave.
>
> The most imortant aspect is that they give rise to tone systems, but
> some aspects are not so clear to me:
>
> 1. Does a scale uniquely distinguish between octave and prime? Most
> representations of scales I have seen, seem to do this
> 2. Why are scales often described as linearly ordered sets? Is it that
> the notion of a cyclic order is not well-known to the corresponding
> theorists, or does it have a deeper philosophical core.
> 3. Are scales anchored on a special position in the tone system as the
> ancient Greek scales or medieval scales or not? Note that the tone
> system itself does not necessarily have to be anchored on a physical
> frequency. For example the discussion about tunings and different
> concert pitches shows that the tone system of a score may be anchored at
> different frequencies.
>
> Finally,
>
> Are scales to be understood as a set of tones or as a sets of chromas?
>
> My intuition is that a scale is a prototypic example of a certain set of
> tones, which has additional structure (order relation, interval group,
> absolute intervals and so on). The student's view is that a scale is the
> set of chromas of the tone system equipped with additional structure
> (order relation, chroma interval group, absolute chroma intervals and so
> on).
>
> When we consider a scale a s a set of tones, it can be also considered
> as rule of the construction of a tone system, while the chroma system
> view suggests, that every single tone of the tone system is already
> somehow (blurred and simplified) contained in the scale.
>
> While in many applications both views have the same consequences, the
> mathematical models differ significantly and have fundamentally
> different properties. The question is which formalisation is better or
> is used and why does the other model not provide the same information.
>
> In order to support my point of view:
> When we want to switch from the chroma view to the tone view (e.g.
> turning an abstract music theoretical idea into a composition) we need a
> prototypic system that allows us to generate and express the missing
> information that is not available to chromas, but is important for
> tones. I think scales can play this role without losing their special
> identity. At least it would be very nice if the music theory already
> knows a structure that can fill this gap.
>
> Yours,
>
> Tobias
>

Charles Lucy
lucy@lucytune.com

-- Promoting global harmony through LucyTuning --

For more information on LucyTuning go to:

http://www.lucytune.com

LucyTuned Lullabies (from around the world) can be found at:

http://www.lullabies.co.uk

Hi Tobias,

I will try to answer your questions from my personal point ov view
and based on my experiences. I think there are no real answers
and maybe my words let come more questions in your mind.

> 1. Does a scale uniquely distinguish between octave and prime?

First of all: there are a lot of scales thar are not based on the
interval on an ocatve (e.g. Bohlen-Pierce, Carlos alpha, beta, gamma).

I developed a nonoctave scale called the Equivocal Tuning that is
an equal division of 2 octaves into 33 steps (more on this you can
find at http://xenharmonic.wikispaces.com/33ed4 http://xenharmonic.wikispaces.com/33ed4).

In this parcicular scale the exact octave lies between step 16 and
step 17. So depending on the melodic and harmonic structure of a
composition, sometimes step 16 plays the role of the octave and
sometimes step 17 does it (for this reason I named the scale equivocal).
This can happen one beneath the other in the same composition.

> 2. Why are scales often described as linearly ordered sets?

Maybe it is the easiest way to describe a scale in ascending order
of its pitches. If you take a look at the left-hand keyboard of an
accordion where the pitches are layed out in fifths you will find it
rather hard to play a melody (but more easy to play I-IV-V cadences).

> 3. Are scales anchored on a special position in the tone system as the
> ancient Greek scales or medieval scales or not?

Music is sung or played on instruments. For singing there is no need
for an absolute pitch to start the scale, but there is a pitch-range
that is comfortable for the singer or singers and the pitch-range

of the composition should fit into the range the singers can sing
(and singing should be fun, too).
Historically, instruments were built on given base-pitches. If you take
a baroque french-horn with no valves then the playable pitches are
predefined by the overtone-series. If you take a violin with 4 strings
tuned in perfect fifths to each other then you have more resonances
and a fuller and richer tone if you play in the pythagorean scale.

Paganini used this by tuning his instrument a half step higher than
the orchestra. So he played in sharp-scales and the orchestra played
in flat-scales (e.g. a D on Paganini's violin was the same pitch as
an E-flat on the violins of his orchestra). So of this Paganini's
violin sounded more full as the violins of the orchestra and his
soli got more volume (today we use microphones and mixing-consoles

so we do not need such tricks any more but for a long time such
things were relevant).
So it seems to me that there are very practical reasons for setting
an absolute pitch being the base of a scale. Most musicians are
really happy to be able to play a composition in the most easiest
scale on their instruments. Only a few instruments like the guitar
allow to play a composition in other keys simply by shifting the
hand-position to another fret without any change in the fingerings.

And if you want your music to be played on real instruments then
it should be rather easy to play. If you program your composition
into a computer or into a barrel-organ so that it will be played
automatically ... I do so because my playing-skills are rather low
and I want to bring the music from my mind into this world.

For my Equivocal Tuning I choose an equal division of the steps
because I designed the scale using a synthesizer that allowed me
to build microtonal scales simply by entering a cent-value for the
step-size of the key-scaling. So I started out at 73 cents because
I only could enter whole numbers. Later on I used Scala do define
my scale and refined the step to 72.72727... cents. On the other hand
an equal division makes modulations to other keys in a composition
much more easier than a pure-tuning. And I love modulations ;-)

In my opinion it is good to know the history of a scale. Having only
12 pitches per octave makes building a keyboard-instrument much more
simpler and cheaper then having 17 or 19 or 31 pitches per octave
(that is one reason why our keyboards look like they do).
For me as a trombone-player an F-sharp is slightly higher than a
G-flat and I use different slide-positions on my trombone for that
2 pitches. For a piano-player or a modern flute-player F-sharp and
G-flat are the same pitch, that means are on the same key or played
using the same fingering.

> Are scales to be understood as a set of tones or as a sets of chromas?

For me, both melodic and harmonic structures are defined on the intervals
between the pitches a given melody or chord is built of. For my
compositions I use one ore more scales as the basic and I add pitches
that do not belong to the scale wherever I feel I need them
(maybe I do not really understand the difference between tones
and chromas).

I hope I could explain my thoughts.

Cheers
Birgit

Hi Tobias,

I will try to answer your questions from my personal point ov view
and based on my experiences. I think there are no real answers
and maybe my words let come more questions in your mind.

> 1. Does a scale uniquely distinguish between octave and prime?

First of all: there are a lot of scales thar are not based on the
interval on an ocatve (e.g. Bohlen-Pierce, Carlos alpha, beta, gamma).

I developed a nonoctave scale called the Equivocal Tuning that is
an equal division of 2 octaves into 33 steps (more on this you can
find at http://xenharmonic.wikispaces.com/33ed4 http://xenharmonic.wikispaces.com/33ed4).

In this parcicular scale the exact octave lies between step 16 and
step 17. So depending on the melodic and harmonic structure of a
composition, sometimes step 16 plays the role of the octave and
sometimes step 17 does it (for this reason I named the scale equivocal).
This can happen one beneath the other in the same composition.

> 2. Why are scales often described as linearly ordered sets?

Maybe it is the easiest way to describe a scale in ascending order
of its pitches. If you take a look at the left-hand keyboard of an
accordion where the pitches are layed out in fifths you will find it
rather hard to play a melody (but more easy to play I-IV-V cadences).

> 3. Are scales anchored on a special position in the tone system as the
> ancient Greek scales or medieval scales or not?

Music is sung or played on instruments. For singing there is no need
for an absolute pitch to start the scale, but there is a pitch-range
that is comfortable for the singer or singers and the pitch-range

of the composition should fit into the range the singers can sing
(and singing should be fun, too).
Historically, instruments were built on given base-pitches. If you take
a baroque french-horn with no valves then the playable pitches are
predefined by the overtone-series. If you take a violin with 4 strings
tuned in perfect fifths to each other then you have more resonances
and a fuller and richer tone if you play in the pythagorean scale.

Paganini used this by tuning his instrument a half step higher than
the orchestra. So he played in sharp-scales and the orchestra played
in flat-scales (e.g. a D on Paganini's violin was the same pitch as
an E-flat on the violins of his orchestra). So of this Paganini's
violin sounded more full as the violins of the orchestra and his
soli got more volume (today we use microphones and mixing-consoles

so we do not need such tricks any more but for a long time such
things were relevant).
So it seems to me that there are very practical reasons for setting
an absolute pitch being the base of a scale. Most musicians are
really happy to be able to play a composition in the most easiest
scale on their instruments. Only a few instruments like the guitar
allow to play a composition in other keys simply by shifting the
hand-position to another fret without any change in the fingerings.

And if you want your music to be played on real instruments then
it should be rather easy to play. If you program your composition
into a computer or into a barrel-organ so that it will be played
automatically ... I do so because my playing-skills are rather low
and I want to bring the music from my mind into this world.

For my Equivocal Tuning I choose an equal division of the steps
because I designed the scale using a synthesizer that allowed me
to build microtonal scales simply by entering a cent-value for the
step-size of the key-scaling. So I started out at 73 cents because
I only could enter whole numbers. Later on I used Scala do define
my scale and refined the step to 72.72727... cents. On the other hand
an equal division makes modulations to other keys in a composition
much more easier than a pure-tuning. And I love modulations ;-)

In my opinion it is good to know the history of a scale. Having only
12 pitches per octave makes building a keyboard-instrument much more
simpler and cheaper then having 17 or 19 or 31 pitches per octave
(that is one reason why our keyboards look like they do).
For me as a trombone-player an F-sharp is slightly higher than a
G-flat and I use different slide-positions on my trombone for that
2 pitches. For a piano-player or a modern flute-player F-sharp and
G-flat are the same pitch, that means are on the same key or played
using the same fingering.

> Are scales to be understood as a set of tones or as a sets of chromas?

For me, both melodic and harmonic structures are defined on the intervals
between the pitches a given melody or chord is built of. For my
compositions I use one ore more scales as the basic and I add pitches
that do not belong to the scale wherever I feel I need them
(maybe I do not really understand the difference between tones
and chromas).

I hope I could explain my thoughts.

Cheers
Birgit

Hi Tobias,

I will try to answer your questions from my personal point ov view
and based on my experiences. I think there are no real answers
and maybe my words let come more questions in your mind.

> 1. Does a scale uniquely distinguish between octave and prime?

First of all: there are a lot of scales thar are not based on the
interval on an ocatve (e.g. Bohlen-Pierce, Carlos alpha, beta, gamma).

I developed a nonoctave scale called the Equivocal Tuning that is
an equal division of 2 octaves into 33 steps (more on this you can
find at http://xenharmonic.wikispaces.com/33ed4 ).

In this parcicular scale the exact octave lies between step 16 and
step 17. So depending on the melodic and harmonic structure of a
composition, sometimes step 16 plays the role of the octave and
sometimes step 17 does it (for this reason I named the scale equivocal).
This can happen one beneath the other in the same composition.

> 2. Why are scales often described as linearly ordered sets?

Maybe it is the easiest way to describe a scale in ascending order
of its pitches. If you take a look at the left-hand keyboard of an
accordion where the pitches are layed out in fifths you will find it
rather hard to play a melody (but more easy to play I-IV-V cadences).

> 3. Are scales anchored on a special position in the tone system as the
> ancient Greek scales or medieval scales or not?

Music is sung or played on instruments. For singing there is no need
for an absolute pitch to start the scale, but there is a pitch-range
that is comfortable for the singer or singers and the pitch-range

of the composition should fit into the range the singers can sing
(and singing should be fun, too).
Historically, instruments were built on given base-pitches. If you take
a baroque french-horn with no valves then the playable pitches are
predefined by the overtone-series. If you take a violin with 4 strings
tuned in perfect fifths to each other then you have more resonances
and a fuller and richer tone if you play in the pythagorean scale.

Paganini used this by tuning his instrument a half step higher than
the orchestra. So he played in sharp-scales and the orchestra played
in flat-scales (e.g. a D on Paganini's violin was the same pitch as
an E-flat on the violins of his orchestra). So of this Paganini's
violin sounded more full as the violins of the orchestra and his
soli got more volume (today we use microphones and mixing-consoles

so we do not need such tricks any more but for a long time such
things were relevant).
So it seems to me that there are very practical reasons for setting
an absolute pitch being the base of a scale. Most musicians are
really happy to be able to play a composition in the most easiest
scale on their instruments. Only a few instruments like the guitar
allow to play a composition in other keys simply by shifting the
hand-position to another fret without any change in the fingerings.

And if you want your music to be played on real instruments then
it should be rather easy to play. If you program your composition
into a computer or into a barrel-organ so that it will be played
automatically ... I do so because my playing-skills are rather low
and I want to bring the music from my mind into this world.

For my Equivocal Tuning I choose an equal division of the steps
because I designed the scale using a synthesizer that allowed me
to build microtonal scales simply by entering a cent-value for the
step-size of the key-scaling. So I started out at 73 cents because
I only could enter whole numbers. Later on I used Scala do define
my scale and refined the step to 72.72727... cents. On the other hand
an equal division makes modulations to other keys in a composition
much more easier than a pure-tuning. And I love modulations ;-)

In my opinion it is good to know the history of a scale. Having only
12 pitches per octave makes building a keyboard-instrument much more

simpler and cheaper then having 17 or 19 or 31 pitches per octave
(that is one reason why our keyboards look like they do).For me as a trombone-player an F-sharp is slightly higher than a
G-flat and I use different slide-positions on my trombone for that
2 pitches. For a piano-player or a modern flute-player F-sharp and
G-flat are the same pitch, that means are on the same key or played
using the same fingering.

> Are scales to be understood as a set of tones or as a sets of chromas?
For me, both melodic and harmonic structures are defined on the intervals

between the pitches a given melody or chord is built of. For my
compositions I use one ore more scales as the basic and I add pitches

that do not belong to the scale wherever I feel I need them
(maybe I do not really understand the difference between tones
and chromas).

I hope I could explain my thoughts.

Cheers
Birgit