User:Lhearne/Extra-Diatonic Intervals: Difference between revisions

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=== Zarlino and Meantone ===

[[File:Della proprietà del numero Senario & della sue parti; & come in esse si ritroua ogni consonanze musicale, figura 1.png|thumb~~|572.986x572.986px~~|''Le institutioni harmoniche,'' Zarlino, 1558, Cap. 15: Della proprietà del numero Senario & della sue parti; & come in esse si ritroua ogni consonanze musicale, figura 1, pg. 25.]]

[[File:Della proprietà del numero Senario & della sue parti; & come in esse si ritroua ogni consonanze musicale, figura 1.png|thumb|''Le institutioni harmoniche,'' Zarlino, 1558, Cap. 15: Della proprietà del numero Senario & della sue parti; & come in esse si ritroua ogni consonanze musicale, figura 1, pg. 25.]]

Intervals were referred to by the Ancient Greek names through the the 18th century, as Latin names. By the Renaissance it had been discovered that a Pythagorean diminished fourth sounded sweet, and approximated the string length ratio 5/4. This just tuning for the major third was sought after, along with the complementary 6/5 tuning for the minor third, and octave complements to both - 8/5 for the minor sixth and 5/3 for the major sixth. Influential Italian music theorist and composer Gioseffo Zarlino put forth that choirs tuned the diatonic scale to the tuning built from this tetrachord, the ''intense diatonic scale'', also known as the ''syntonic or syntonus diatonic scale'' or the ''Ptolemaic sequence'':

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In addition to the Latin interval names, derived from the Ancient Greek interval names, we see on the diagram a single interval name in Italian: ''Essachordo maggiore'', referring to the ratio 5/3, which we are tempted to translate to 'major sixth'. Chapter 16, ''Quel che sia Consonanze semplice, e Composta; & che nel Senario si ritouano le sorme di tutte le somplici consonanze; & onde habbia origine l'Essachordo minore'', puts forward that the ''Essachordo minore,'' or perhaps 'minor sixth' be tuned to 8/5.

In the 1691 ''Lettre de Monsieur Huygens à l'Auteur [Henri Basnage de Beauval] touchant le Cycle Harmonique,'' theorist Christiaan Huygens gave names and ratios to common intervals and mapped them to 31-tET, which very closely approximates 1/4-comma Meantone. Translated from French, 3/2 was labelled a Fifth, 4/3 a Fourth, 5/4 a major Third, 6/5 and minor Third, 5/3 a major Sixth and 8/5 a minor Sixth. Here we really begin to see today's interval names.

=== English interval names in the Baroque ===

[[File:Harmonics, or The Philosophy of Musical Sounds, Section 2 figure 3.png|thumb~~|547.986x547.986px~~|''Harmonics, or The Philosophy of Musical Sounds'', Edition 2, Smith, 1759, Section 2: On the Names and Notation of consonance and their intervals, Fig. 2 & 3 , pg. 10]]

[[File:Harmonics, or The Philosophy of Musical Sounds, Section 2 figure 3.png|thumb|''Harmonics, or The Philosophy of Musical Sounds'', Edition 2, Smith, 1759, Section 2: On the Names and Notation of consonance and their intervals, Fig. 2 & 3 , pg. 10]]

After English superseded Latin as the the main language of scholarship, the Latin interval names were rejected and the convention we saw in Zarlino's Italian for naming the smaller of a pair of sizes of an interval 'minor' and the larger 'major' was further applied.

Music theorist and mathematician Robert Smith provides the diagram and table on the right in his 1749 ''Harmonics, or, The Philosophy of Musical Sounds,'' with the description: <blockquote>'Fig. 2. If a musical string ''CO'' and it's parts ''DO'', ''EO'', ''FO'', ''GO'', ''AO'', ''BO'', ''cO'', be in proportion to one another as the numbers 1, 8/9, 4/5, 3/4, 2/3, 3/5, 8/15, 1/2, their vibrations will exhibit the system of 8 sounds which musicians donate by the letters ''C'', ''D'', ''E'', ''F'', ''G'', ''A'', ''B'', ''c''.</blockquote><blockquote>Fig. 3. And supposing those strings to be ranged like ordinates to a right line ''Cc'', and their distances ''CD'', ''DE'', ''EF'', ''FG'', ''GA'', ''AB'', ''BC'', not to be the differences of their lengths, as in fig. 2. but to be the magnitudes proportional to the intervals of their sounds, the received Names of these intervals are shewn in the following Table; and are taken from the numbers of the strings or sounds in each interval inclusively; as a Second, Third, Fourth, Fifth, &c, with the epithet of ''major'' or ''minor'', according as the name or number belongs to a greater of smaller total interval; the difference of which results chiefly from the different magnitudes of the major and minor second, called the Tone and Hemitone.'</blockquote>We note that Smith uses the tuning of the diatonic scale that Zarlino put forward: the Ptolemaic Sequence.

We may be surprised to see 4:3 here labelled as a minor Fourth, and 3/2 as a major Fifth, but it is obvious that this naming is more consistent than today's. Smith adds that <blockquote>'Any one of the ratios in the third column of the foregoing Table, except 80 to 81, or any one of them compounded once of oftener with the ratio 2 to 1 or 1 to 2, is called a Perfect ratio when reduced to it's least terms. And when the times of the single vibrations of any two sounds have a perfect ratio, the consonance and it's interval too after called Perfect; and is called Imperfect or Tempered when that perfect ratio and interval is a little increased or decreased.'</blockquote><blockquote>'Any small increment of decrement of a perfect interval is called respectively the Sharp or Flat Temperament of the imperfect consonance, and is measured most conveniently by the proportion it bears to the comma'</blockquote>Therefore in this system 3/2 is the ''Perfect major Fifth'' and 5/4 the ''Perfect major Third''. 81/64 might be labelled a ''comma sharp major Third'', 32/27 a ''comma flat minor Third'', and the 1/4-comma Meantone fifth a ''1/4-comma flat major Fifth''. The interval naming scheme Smith describes may be immediately applied to 5-limit microtonal systems. There is an inconsistency, however, where It seems that 9/8 should be called a ''Perfect major Second,'' but that, while 9/5 be named a ''comma sharp minor Seventh'', it's inverse, 10/9, is a ''Perfect minor Tone.''

We may be surprised to see 4:3 here labelled as a minor Fourth, and 3/2 as a major Fifth, but it is obvious that this naming is more consistent than today's. Smith adds that <blockquote>'Any one of the ratios in the third column of the foregoing Table, except 80 to 81, or any one of them compounded once of oftener with the ratio 2 to 1 or 1 to 2, is called a Perfect ratio when reduced to it's least terms. And when the times of the single vibrations of any two sounds have a perfect ratio, the consonance and it's interval too after called Perfect; and is called Imperfect or Tempered when that perfect ratio and interval is a little increased or decreased.'</blockquote><blockquote>'Any small increment of decrement of a perfect interval is called respectively the Sharp or Flat Temperament of the imperfect consonance, and is measured most conveniently by the proportion it bears to the comma'</blockquote>Therefore in this system 3/2 is the ''Perfect major Fifth'' and 5/4 the ''Perfect major Third''. 81/64 might be labelled a ''comma sharp major Third'', 32/27 a ''comma flat minor Third'', and the 1/4-comma Meantone fifth a ''1/4-comma flat major Fifth''. The interval naming scheme Smith describes may be immediately applied to 5-limit microtonal systems. There is an inconsistency, however, where it seems that 9/8 should be called a ''Perfect major Second,'' but that, while 9/5 be named a ''comma sharp minor Seventh'', it's inverse, 10/9, is a ''Perfect minor Tone.''

[[File:Helmholtz consonances table.png|thumb~~|616.997x616.997px~~|Table describing the influence of the different consonances on one another, up to the 9th partial, from ''On the Sensations of Tone as a Phycological Basis for Theory of Music'', Helmholtz, 1863, Translation by Ellis, 1875, pg. 187]]

In ''An Elementary Treatise on Musical Intervals and Temperament,'' published in 1876'','' R. H. M Bosanquet refers to 5/4 as the ''perfect third'', and 81/64 as the P''ythagorean third''. Bosanquet also labels other intervals of the Pythagorean diatonic scale similarly, i.e. 256/243, the limma, is labelled the ''Pythagorean semitone'', and 27/16 the ''Pythagorean sixth''. 81/80 is labelled the ''ordinary comma'', or simple the ''comma,'' and the Pythagorean comma is defined as the difference between twelve fifths and seven octaves. The apotome of 2187/2048 is referred to as Apatomè Pythagoria. The following relationships are then described:

[[File:Helmholtz consonances table.png|thumb|Table describing the influence of the different consonances on one another, up to the 9th partial, from ''On the Sensations of Tone as a Phycological Basis for Theory of Music'', Helmholtz, 1863, Translation by Ellis, 1875, pg. 187]]

=== Helmholtz and Ellis ===

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Each interval name has two sizes that differ by the comma 242:243. The notation included in the table is from HEWM notation, developed as an extension to the Helmholtz-Ellis use of '+' and '-' by Joe Monzo (http://www.tonalsoft.com/enc/h/hewm.aspx<nowiki/>).'^' indicates raising 'v' a lowered of 33/32. In HEWM notation '+' and '-' are refined to mean raising and lowering of 81/80 respectively and '>' and '<' are added instead to indicate raising and lowering of 64/63. Letter names correspond instead of the the Ptolemaic sequence, as in Smith's and Helmholtz' descriptions, but to a Pythagorean tuning of the diatonic scale, where '#' and '♭' and respectively raise and lower the apotome, 2187/2048. HEWM notation is not accompanied by an interval naming system.

=== Common microtonal interval ~~names~~ ===

=== Common interval names today ===

~~This system was~~ further generalised by some theorists and musicians such that an interval a bit smaller than a major is referred to as a ''subminor third'', and an interval a bit larger than a minor third as a ''supraminor third''. Notice that 'supra' is used instead of 'super', but 'sub' is still used. Similarly defined are submajor and supraminor seconds, sixths and sevenths. 'Sub' and 'super' prefixes have also seen occasional application to the perfect scale degrees. Like in the case of the submajor 3rd, etc., super and sub unisons, fourths, fifths and octaves are not associated with particular frequency ratio by all or most microtonal musicians and theorists. In the case of seconds, thirds, sixths and sevenths, intervals half way between major and minor are often called ''neutral''. Finally, in limited use are 'intermediates', where an interval in-between a major 3rd and a perfect fourth, for example, is referred to as a 'third-fourth'. There are undoubtedly other interval naming practice that exists that are not as well known to the author, which are likely to be less commonly used. Although all these microtonal interval naming concepts are in common use, there is not yet a complete system that defined them, only complete systems that depart from them.

These interval names are used by theorists and microtonal musicians today, though 7:5 and 10:7 are given many different names, today also considered to be an augmented fourth and diminished fifth, lesser septimal tritone and greater septimal tritone, or simply as tritones. The fourth and fifth are today called perfect fourth and perfect fifth, and Smith's major Fourth and minor Fifth referred to as augmented fourth and diminished fifth respectively. As can be seen in Tchaikovsky's ''A Guide to the Practical Study of Harmony,'' by the beginning of end of the 19th century the familiar conventions for the naming of intervals were set, wherein

* Seconds, thirds, sixths and sevenths appear in the diatonic in two sizes, the larger labelled 'major' and the smaller, 'minor'.

* Major, when raised by a semitone, becomes 'augmented', and minor, lowered by a semitone, 'diminished'.

* The smaller of the two sizes of fourth and the larger of the two sizes of fifth are labelled 'perfect', along with the unison and octave.

* A perfect interval, when raised a semitone is labelled 'augmented', and when lowered a semitone, 'diminished'.

=== Extended Meantone ===

Super and sub were further generalised by some theorists and musicians such that an interval a bit smaller than a major is referred to as a ''subminor third'', and an interval a bit larger than a minor third as a ''supraminor third''.

Notice that 'supra' is used instead of 'super', but 'sub' is still used. Finally, in limited use are 'intermediates', where an interval in-between a major 3rd and a perfect fourth, for example, is referred to as a 'third-fourth'. There are undoubtedly other interval naming practice that exists that are not as well known to the author, which are likely to be less commonly used. Although all these microtonal interval naming concepts are in common use, there is not yet a complete system that defined them, only complete systems that depart from them.

=== [[Sagittal notation|Sagittal]] - [http://forum.sagittal.org/viewforum.php?f=9 sagispeak] ===

@@ Line 33: / Line 33: @@
 === Zarlino and Meantone ===
-[[File:Della proprietà del numero Senario & della sue parti; & come in esse si ritroua ogni consonanze musicale, figura 1.png|thumb|572.986x572.986px|''Le institutioni harmoniche,'' Zarlino, 1558, Cap. 15: Della proprietà del numero Senario & della sue parti; & come in esse si ritroua ogni consonanze musicale, figura 1, pg. 25.]]
+[[File:Della proprietà del numero Senario & della sue parti; & come in esse si ritroua ogni consonanze musicale, figura 1.png|thumb|''Le institutioni harmoniche,'' Zarlino, 1558, Cap. 15: Della proprietà del numero Senario & della sue parti; & come in esse si ritroua ogni consonanze musicale, figura 1, pg. 25.]]
 Intervals were referred to by the Ancient Greek names through the the 18th century, as Latin names. By the Renaissance it had been discovered that a Pythagorean diminished fourth sounded sweet, and approximated the string length ratio 5/4. This just tuning for the major third was sought after, along with the complementary 6/5 tuning for the minor third, and octave complements to both - 8/5 for the minor sixth and 5/3 for the major sixth. Influential Italian music theorist and composer Gioseffo Zarlino put forth that choirs tuned the diatonic scale to the tuning built from this tetrachord, the ''intense diatonic scale'', also known as the ''syntonic or syntonus diatonic scale'' or the ''Ptolemaic sequence'':
@@ Line 43: / Line 43: @@
 In addition to the Latin interval names, derived from the Ancient Greek interval names, we see on the diagram a single interval name in Italian: ''Essachordo maggiore'', referring to the ratio 5/3, which we are tempted to translate to 'major sixth'. Chapter 16, ''Quel che sia Consonanze semplice, e Composta; & che nel Senario si ritouano le sorme di tutte le somplici consonanze; & onde habbia origine l'Essachordo minore'', puts forward that the ''Essachordo minore,'' or perhaps 'minor sixth' be tuned to 8/5.
+In the 1691 ''Lettre de Monsieur Huygens à l'Auteur [Henri Basnage de Beauval] touchant le Cycle Harmonique,'' theorist Christiaan Huygens gave names and ratios to common intervals and mapped them to 31-tET, which very closely approximates 1/4-comma Meantone. Translated from French, 3/2 was labelled a Fifth, 4/3 a Fourth, 5/4 a major Third, 6/5 and minor Third, 5/3 a major Sixth and 8/5 a minor Sixth. Here we really begin to see today's interval names.
 === English interval names in the Baroque ===
-[[File:Harmonics, or The Philosophy of Musical Sounds, Section 2 figure 3.png|thumb|547.986x547.986px|''Harmonics, or The Philosophy of Musical Sounds'', Edition 2, Smith, 1759, Section 2: On the Names and Notation of consonance and their intervals, Fig. 2 & 3 , pg. 10]]
+[[File:Harmonics, or The Philosophy of Musical Sounds, Section 2 figure 3.png|thumb|''Harmonics, or The Philosophy of Musical Sounds'', Edition 2, Smith, 1759, Section 2: On the Names and Notation of consonance and their intervals, Fig. 2 & 3 , pg. 10]]
 After English superseded Latin as the the main language of scholarship, the Latin interval names were rejected and the convention we saw in Zarlino's Italian for naming the smaller of a pair of sizes of an interval 'minor' and the larger 'major' was further applied.
 Music theorist and mathematician Robert Smith provides the diagram and table on the right in his 1749 ''Harmonics, or, The Philosophy of Musical Sounds,'' with the description: <blockquote>'Fig. 2. If a musical string ''CO'' and it's parts ''DO'', ''EO'', ''FO'', ''GO'', ''AO'', ''BO'', ''cO'', be in proportion to one another as the numbers 1, 8/9, 4/5, 3/4, 2/3, 3/5, 8/15, 1/2, their vibrations will exhibit the system of 8 sounds which musicians donate by the letters ''C'', ''D'', ''E'', ''F'', ''G'', ''A'', ''B'', ''c''.</blockquote><blockquote>Fig. 3. And supposing those strings to be ranged like ordinates to a right line ''Cc'', and their distances ''CD'', ''DE'', ''EF'', ''FG'', ''GA'', ''AB'', ''BC'', not to be the differences of their lengths, as in fig. 2. but to be the magnitudes proportional to the intervals of their sounds, the received Names of these intervals are shewn in the following Table; and are taken from the numbers of the strings or sounds in each interval inclusively; as a Second, Third, Fourth, Fifth, &c, with the epithet of ''major'' or ''minor'', according as the name or number belongs to a greater of smaller total interval; the difference of which results chiefly from the different magnitudes of the major and minor second, called the Tone and Hemitone.'</blockquote>We note that Smith uses the tuning of the diatonic scale that Zarlino put forward: the Ptolemaic Sequence.
-We may be surprised to see 4:3 here labelled as a minor Fourth, and 3/2 as a major Fifth, but it is obvious that this naming is more consistent than today's. Smith adds that <blockquote>'Any one of the ratios in the third column of the foregoing Table, except 80 to 81, or any one of them compounded once of oftener with the ratio 2 to 1 or 1 to 2, is called a Perfect ratio when reduced to it's least terms. And when the times of the single vibrations of any two sounds have a perfect ratio, the consonance and it's interval too after called Perfect; and is called Imperfect or Tempered when that perfect ratio and interval is a little increased or decreased.'</blockquote><blockquote>'Any small increment of decrement of a perfect interval is called respectively the Sharp or Flat Temperament of the imperfect consonance, and is measured most conveniently by the proportion it bears to the comma'</blockquote>Therefore in this system 3/2 is the ''Perfect major Fifth'' and 5/4 the ''Perfect major Third''. 81/64 might be labelled a ''comma sharp major Third'', 32/27 a ''comma flat minor Third'', and the 1/4-comma Meantone fifth a ''1/4-comma flat major Fifth''. The interval naming scheme Smith describes may be immediately applied to 5-limit microtonal systems. There is an inconsistency, however, where It seems that 9/8 should be called a ''Perfect major Second,'' but that, while 9/5 be named a ''comma sharp minor Seventh'', it's inverse, 10/9, is a ''Perfect minor Tone.''
+We may be surprised to see 4:3 here labelled as a minor Fourth, and 3/2 as a major Fifth, but it is obvious that this naming is more consistent than today's. Smith adds that <blockquote>'Any one of the ratios in the third column of the foregoing Table, except 80 to 81, or any one of them compounded once of oftener with the ratio 2 to 1 or 1 to 2, is called a Perfect ratio when reduced to it's least terms. And when the times of the single vibrations of any two sounds have a perfect ratio, the consonance and it's interval too after called Perfect; and is called Imperfect or Tempered when that perfect ratio and interval is a little increased or decreased.'</blockquote><blockquote>'Any small increment of decrement of a perfect interval is called respectively the Sharp or Flat Temperament of the imperfect consonance, and is measured most conveniently by the proportion it bears to the comma'</blockquote>Therefore in this system 3/2 is the ''Perfect major Fifth'' and 5/4 the ''Perfect major Third''. 81/64 might be labelled a ''comma sharp major Third'', 32/27 a ''comma flat minor Third'', and the 1/4-comma Meantone fifth a ''1/4-comma flat major Fifth''. The interval naming scheme Smith describes may be immediately applied to 5-limit microtonal systems. There is an inconsistency, however, where it seems that 9/8 should be called a ''Perfect major Second,'' but that, while 9/5 be named a ''comma sharp minor Seventh'', it's inverse, 10/9, is a ''Perfect minor Tone.''
-[[File:Helmholtz consonances table.png|thumb|616.997x616.997px|Table describing the influence of the different consonances on one another, up to the 9th partial, from ''On the Sensations of Tone as a Phycological Basis for Theory of Music'', Helmholtz, 1863, Translation by Ellis, 1875, pg. 187]]
+In ''An Elementary Treatise on Musical Intervals and Temperament,'' published in 1876'','' R. H. M Bosanquet refers to 5/4 as the ''perfect third'', and 81/64 as the P''ythagorean third''. Bosanquet also labels other intervals of the Pythagorean diatonic scale similarly, i.e. 256/243, the limma, is labelled the ''Pythagorean semitone'', and 27/16 the ''Pythagorean sixth''. 81/80 is labelled the ''ordinary comma'', or simple the ''comma,'' and the Pythagorean comma is defined as the difference between twelve fifths and seven octaves. The apotome of 2187/2048 is referred to as Apatomè Pythagoria. The following relationships are then described:
+[[File:Helmholtz consonances table.png|thumb|Table describing the influence of the different consonances on one another, up to the 9th partial, from ''On the Sensations of Tone as a Phycological Basis for Theory of Music'', Helmholtz, 1863, Translation by Ellis, 1875, pg. 187]]
 === Helmholtz and Ellis ===
@@ Line 152: / Line 157: @@
 Each interval name has two sizes that differ by the comma 242:243. The notation included in the table is from HEWM notation, developed as an extension to the Helmholtz-Ellis use of '+' and '-' by Joe Monzo (http://www.tonalsoft.com/enc/h/hewm.aspx<nowiki/>).'^' indicates raising 'v' a lowered of 33/32. In HEWM notation '+' and '-' are refined to mean raising and lowering of 81/80 respectively and '>' and '<' are added instead to indicate raising and lowering of 64/63. Letter names correspond instead of the the Ptolemaic sequence, as in Smith's and Helmholtz' descriptions, but to a Pythagorean tuning of the diatonic scale, where '#' and '♭' and respectively raise and lower the apotome, 2187/2048. HEWM notation is not accompanied by an interval naming system.
-=== Common microtonal interval names ===
+=== Common interval names today ===
-This system was further generalised by some theorists and musicians such that an interval a bit smaller than a major is referred to as a ''subminor third'', and an interval a bit larger than a minor third as a ''supraminor third''. Notice that 'supra' is used instead of 'super', but 'sub' is still used. Similarly defined are submajor and supraminor seconds, sixths and sevenths. 'Sub' and 'super' prefixes have also seen occasional application to the perfect scale degrees. Like in the case of the submajor 3rd, etc., super and sub unisons, fourths, fifths and octaves are not associated with particular frequency ratio by all or most microtonal musicians and theorists. In the case of seconds, thirds, sixths and sevenths, intervals half way between major and minor are often called ''neutral''. Finally, in limited use are 'intermediates', where an interval in-between a major 3rd and a perfect fourth, for example, is referred to as a 'third-fourth'. There are undoubtedly other interval naming practice that exists that are not as well known to the author, which are likely to be less commonly used. Although all these microtonal interval naming concepts are in common use, there is not yet a complete system that defined them, only complete systems that depart from them.
+These interval names are used by theorists and microtonal musicians today, though 7:5 and 10:7 are given many different names, today also considered to be an augmented fourth and diminished fifth, lesser septimal tritone and greater septimal tritone, or simply as tritones. The fourth and fifth are today called perfect fourth and perfect fifth, and Smith's major Fourth and minor Fifth referred to as augmented fourth and diminished fifth respectively. As can be seen in Tchaikovsky's ''A Guide to the Practical Study of Harmony,'' by the beginning of end of the 19th century the familiar conventions for the naming of intervals were set, wherein
+* Seconds, thirds, sixths and sevenths appear in the diatonic in two sizes, the larger labelled 'major' and the smaller, 'minor'.
+* Major, when raised by a semitone, becomes 'augmented', and minor, lowered by a semitone, 'diminished'.
+* The smaller of the two sizes of fourth and the larger of the two sizes of fifth are labelled 'perfect', along with the unison and octave.
+* A perfect interval, when raised a semitone is labelled 'augmented', and when lowered a semitone, 'diminished'.
+=== Extended Meantone ===
+Super and sub were further generalised by some theorists and musicians such that an interval a bit smaller than a major is referred to as a ''subminor third'', and an interval a bit larger than a minor third as a ''supraminor third''.
+Notice that 'supra' is used instead of 'super', but 'sub' is still used. Finally, in limited use are 'intermediates', where an interval in-between a major 3rd and a perfect fourth, for example, is referred to as a 'third-fourth'. There are undoubtedly other interval naming practice that exists that are not as well known to the author, which are likely to be less commonly used. Although all these microtonal interval naming concepts are in common use, there is not yet a complete system that defined them, only complete systems that depart from them.
 === [[Sagittal notation|Sagittal]] - [http://forum.sagittal.org/viewforum.php?f=9 sagispeak] ===