Introductory examples in Sagittal notation: Difference between revisions

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This page lists a few elementary examples that hopefully shed some light on the philosophy (and the pitfalls!) of [[Sagittal_notation|Sagittal notation]]. For a detailed introduction into Sagittal notation the document [[:File:Sagittal.pdf|Sagittal.pdf]] is the reference.

=Just intonation: notating ~~an overtone~~ scale=

=Just intonation: notating a harmonic scale=

As the introduction [[:File:Sagittal.pdf|Sagittal.pdf]] says, Sagittal notation uses a conventional staff on which the natural notes are in a single series of fifths, i.e, in the case of just intonation, in [[3-limit|Pythagorean tuning]]. The meaning of conventional sharps and flats is Pythagorean as well, i.e. they stand for raising the corresponding note by a [[2187/2048|Pythagorean chromatic semitone (apotome)]], a "large" semitone 113.7 cents in size.

For the notation of notes in higher [[Harmonic_Limit|limits]], additional symbols are introduced. The intervals these symbols stand for are mostly [[Comma|commas]] - the maybe most elementary example is the [[81/80|syntonic comma]] (ratio 81/80, 21.506 cents), the difference between a pythagorean and a just major third, necessary for just intonation in [[5-limit|5-limit]]. Other elementary commas appearing along the ~~overtone~~ series are: in [[7-limit|7-limit]] the [[64/63|septimal comma or Architas' comma]] (64/63, 27.264 Cents), the difference between a minor and a harmonic seventh, and, in [[11-limit|11-limit]], the [[33/32|undecimal comma or al-Farabi quarter-tone]] (33/32, 53.2729 cents), the difference between an undecimal semi-augmented and a perfect fourth.

For the notation of notes in higher [[Harmonic_Limit|limits]], additional symbols are introduced. The intervals these symbols stand for are mostly [[Comma|commas]] - the maybe most elementary example is the [[81/80|syntonic comma]] (ratio 81/80, 21.506 cents), the difference between a pythagorean and a just major third, necessary for just intonation in [[5-limit|5-limit]]. Other elementary commas appearing along the harmonic series are: in [[7-limit|7-limit]] the [[64/63|septimal comma or Architas' comma]] (64/63, 27.264 Cents), the difference between a minor and a harmonic seventh, and, in [[11-limit|11-limit]], the [[33/32|undecimal comma or al-Farabi quarter-tone]] (33/32, 53.2729 cents), the difference between an undecimal semi-augmented and a perfect fourth.

With the Sagittal symbos for these three commas, a scale consisting of the ~~overtones~~ 4 to 11 can be written as follows:

With the Sagittal symbos for these three commas, a scale consisting of the harmonics 4 to 11 can be written as follows:

[[File:SagittalOvertoneSeries.jpg|alt=SagittalOvertoneSeries.jpg|SagittalOvertoneSeries.jpg]]

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[[File:SagittalEDOExample.jpg|alt=SagittalEDOExample.jpg|SagittalEDOExample.jpg]]

What is displayed is the tetrad consisting of the ~~overtones~~ 4 to 7 (respectively, in equal temperaments, of their approximations), a chord resembling a dominant seventh chord. It appears first in mixed Sagittal notation for just intonation, with the syntonic comma symbol at the E note and the septimal comma symbol at the Bb note.

What is displayed is the tetrad consisting of the harmonics 4 to 7 (respectively, in equal temperaments, of their approximations), a chord resembling a dominant seventh chord. It appears first in mixed Sagittal notation for just intonation, with the syntonic comma symbol at the E note and the septimal comma symbol at the Bb note.

In equal temperaments, we could, in theory, write it always like that, following guideline 1. But depending on the concrete equal division there will be harmonic equivalences that suggest certain simplifications.

@@ Line 2: / Line 2: @@
 This page lists a few elementary examples that hopefully shed some light on the philosophy (and the pitfalls!) of [[Sagittal_notation|Sagittal notation]]. For a detailed introduction into Sagittal notation the document [[:File:Sagittal.pdf|Sagittal.pdf]] is the reference.
-=Just intonation: notating an overtone scale=
+=Just intonation: notating a harmonic scale=
 As the introduction [[:File:Sagittal.pdf|Sagittal.pdf]] says, Sagittal notation uses a conventional staff on which the natural notes are in a single series of fifths, i.e, in the case of just intonation, in [[3-limit|Pythagorean tuning]]. The meaning of conventional sharps and flats is Pythagorean as well, i.e. they stand for raising the corresponding note by a [[2187/2048|Pythagorean chromatic semitone (apotome)]], a "large" semitone 113.7 cents in size.
-For the notation of notes in higher [[Harmonic_Limit|limits]], additional symbols are introduced. The intervals these symbols stand for are mostly [[Comma|commas]] - the maybe most elementary example is the [[81/80|syntonic comma]] (ratio 81/80, 21.506 cents), the difference between a pythagorean and a just major third, necessary for just intonation in [[5-limit|5-limit]]. Other elementary commas appearing along the overtone series are: in [[7-limit|7-limit]] the [[64/63|septimal comma or Architas' comma]] (64/63, 27.264 Cents), the difference between a minor and a harmonic seventh, and, in [[11-limit|11-limit]], the [[33/32|undecimal comma or al-Farabi quarter-tone]] (33/32, 53.2729 cents), the difference between an undecimal semi-augmented and a perfect fourth.
+For the notation of notes in higher [[Harmonic_Limit|limits]], additional symbols are introduced. The intervals these symbols stand for are mostly [[Comma|commas]] - the maybe most elementary example is the [[81/80|syntonic comma]] (ratio 81/80, 21.506 cents), the difference between a pythagorean and a just major third, necessary for just intonation in [[5-limit|5-limit]]. Other elementary commas appearing along the harmonic series are: in [[7-limit|7-limit]] the [[64/63|septimal comma or Architas' comma]] (64/63, 27.264 Cents), the difference between a minor and a harmonic seventh, and, in [[11-limit|11-limit]], the [[33/32|undecimal comma or al-Farabi quarter-tone]] (33/32, 53.2729 cents), the difference between an undecimal semi-augmented and a perfect fourth.
-With the Sagittal symbos for these three commas, a scale consisting of the overtones 4 to 11 can be written as follows:
+With the Sagittal symbos for these three commas, a scale consisting of the harmonics 4 to 11 can be written as follows:
 [[File:SagittalOvertoneSeries.jpg|alt=SagittalOvertoneSeries.jpg|SagittalOvertoneSeries.jpg]]
@@ Line 24: / Line 24: @@
 [[File:SagittalEDOExample.jpg|alt=SagittalEDOExample.jpg|SagittalEDOExample.jpg]]
-What is displayed is the tetrad consisting of the overtones 4 to 7 (respectively, in equal temperaments, of their approximations), a chord resembling a dominant seventh chord. It appears first in mixed Sagittal notation for just intonation, with the syntonic comma symbol at the E note and the septimal comma symbol at the Bb note.
+What is displayed is the tetrad consisting of the harmonics 4 to 7 (respectively, in equal temperaments, of their approximations), a chord resembling a dominant seventh chord. It appears first in mixed Sagittal notation for just intonation, with the syntonic comma symbol at the E note and the septimal comma symbol at the Bb note.
 In equal temperaments, we could, in theory, write it always like that, following guideline 1. But depending on the concrete equal division there will be harmonic equivalences that suggest certain simplifications.