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Kite's Genchain mode numbering: Difference between revisions

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__FORCETOC__
=MOS Scales=
=MOS Scales=
__FORCETOC__
 
'''Mode Numbers''' provide a way to name MOS, MODMOS and even non-MOS rank-2 scales and modes systematically. Like [[Modal_UDP_Notation|Modal UDP notation]], it starts with the convention of using ''some-temperament-name'' [''some-number''] to create a generator-chain, and adds a way to number each mode uniquely.
'''Mode Numbers''' provide a way to name MOS, MODMOS and even non-MOS rank-2 scales and modes systematically. Like [[Modal_UDP_Notation|Modal UDP notation]], it starts with the convention of using ''some-temperament-name'' [''some-number''] to create a generator-chain, and adds a way to number each mode uniquely.


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{| class="wikitable"
{| class="wikitable"
|-
|-
| | old scale name
! | old scale name
| | new scale name
! | new scale name
| | sL pattern
! | sL pattern
| | example on white keys
! | example on white keys
| | genchain
! | genchain
|-
|-
| | Lydian
| | Lydian
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{| class="wikitable"
{| class="wikitable"
|-
|-
| | old scale name
! | old scale name
| | new scale name
! | new scale name
| | sL pattern
! | sL pattern
| | example in C
! | example in C
| | ------------------- genchain ---------------
! | ------------------- genchain ---------------
|-
|-
| | Lydian
| | Lydian
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{| class="wikitable"
{| class="wikitable"
|-
|-
| | old scale name
! | old scale name
| | new scale name
! | new scale name
| | sL pattern
! | sL pattern
| | example in C
! | example in C
| | --------- genchain -------
! | --------- genchain -------
|-
|-
| | major pentatonic
| | major pentatonic
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{| class="wikitable"
{| class="wikitable"
|-
|-
| | scale name
! | scale name
| | sL pattern (assumes
! | sL pattern (assumes<br>~3/2 &lt; 700¢)
 
! | example in C
~3/2 &lt; 700¢)
! | genchain
| | example in C
| | genchain
|-
|-
| | 1st Meantone [12]
| | 1st Meantone [12]
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If the fifth were larger than 700¢, which would be the case for Superpyth[12], L and s would be interchanged.
If the fifth were larger than 700¢, which would be the case for Superpyth[12], L and s would be interchanged.


[[Sensi|Sensi]] [8] modes in 19edo (generator = 3rd = ~9/7 = 7\19, L = 3\19, s = 2\19)
[[Sensi]] [8] modes in 19edo (generator = 3rd = ~9/7 = 7\19, L = 3\19, s = 2\19)


{| class="wikitable"
{| class="wikitable"
|-
|-
| | scale name
! | scale name
| | sL pattern
! | sL pattern
| | example in C
! | example in C
| | genchain
! | genchain
|-
|-
| | 1st Sensi [8]
| | 1st Sensi [8]
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1st Sensi[8] would be C D E F G Hb A B C, 2nd would be C D E F G H A B C, etc.
1st Sensi[8] would be C D E F G Hb A B C, 2nd would be C D E F G H A B C, etc.


[[Porcupine|Porcupine]] [7] modes in 22edo (generator = 2nd = ~10/9 = 3\22, L = 4\22, s = 3\22), using [[Ups_and_Downs_Notation|ups and downs notation]]. Because the generator is a 2nd, the genchain resembles the scale.
[[Porcupine]] [7] modes in 22edo (generator = 2nd = ~10/9 = 3\22, L = 4\22, s = 3\22), using [[ups and downs notation]]. Because the generator is a 2nd, the genchain resembles the scale.


{| class="wikitable"
{| class="wikitable"
|-
|-
| | scale name
! | scale name
| | sL pattern
! | sL pattern
| | example in C
! | example in C
| | genchain
! | genchain
|-
|-
| | 1st Porcupine [7]
| | 1st Porcupine [7]
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C 1st Porcupine [7] would be C D E F G Ab Bb C, 2nd would be C D E F G Ab B C, etc.
C 1st Porcupine [7] would be C D E F G Ab Bb C, 2nd would be C D E F G Ab B C, etc.


='''<u>MODMOS scales</u>'''=
=MODMOS scales=


[[MODMOS_Scales|MODMOS]] scales are named as chromatic alterations of a MOS scale, similar to UDP notation. The ascending melodic minor scale is 5th Meantone [7] #6 #7. The "#" symbol means moved N steps forwards on the genchain, whether the generator is chroma-positive or not. This scale has the same name in 16edo, even though in 16edo, G# is actually flat of G. A good alternative, especially for non-heptatonic and non-fifth-based scales, is to use + and - for forwards and backwards, as in 5th Meantone [7] +6 +7.
[[MODMOS scales]] are named as chromatic alterations of a MOS scale, similar to UDP notation. The ascending melodic minor scale is 5th Meantone [7] #6 #7. The "#" symbol means moved N steps forwards on the genchain, whether the generator is chroma-positive or not. This scale has the same name in 16edo, even though in 16edo, G# is actually flat of G. A good alternative, especially for non-heptatonic and non-fifth-based scales, is to use + and - for forwards and backwards, as in 5th Meantone [7] +6 +7.


MODMOS names are ambiguous. The ascending melodic minor scale could also be written as 2nd Meantone [7] b3 (major scale with a minor 3rd), or as 4th Meantone [7] #7 (dorian with a major 7th).
MODMOS names are ambiguous. The ascending melodic minor scale could also be written as 2nd Meantone [7] b3 (major scale with a minor 3rd), or as 4th Meantone [7] #7 (dorian with a major 7th).
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{| class="wikitable"
{| class="wikitable"
|-
|-
| | old scale name
! | old scale name
| | example in A
! | example in A
| | genchain
! | genchain
| | new scale name
! | new scale name
| | sML pattern
! | sML pattern
|-
|-
| | Harmonic minor
| | Harmonic minor
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7th Meantone [7] #2: C D Eb F Gb Ab Bb C
7th Meantone [7] #2: C D Eb F Gb Ab Bb C


='''<u>Fractional-octave periods</u>'''=
=Fractional-octave periods=


Fractional-period rank-2 temperaments have multiple genchains running in parallel. Multiple genchains occur because a rank-2 genchain is really a 2 dimensional "genweb", running in octaves (or whatever the period is) vertically and fifths (or whatever the generator is) horizontally.
Fractional-period rank-2 temperaments have multiple genchains running in parallel. Multiple genchains occur because a rank-2 genchain is really a 2 dimensional "genweb", running in octaves (or whatever the period is) vertically and fifths (or whatever the generator is) horizontally.
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{| class="wikitable"
{| class="wikitable"
|-
|-
| | scale name
! | scale name
| | sL pattern
! | sL pattern
| | example in C
! | example in C
| | 1st genchain
! | 1st genchain
| | 2nd genchain
! | 2nd genchain
|-
|-
| | 1st Srutal [10]
| | 1st Srutal [10]
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{| class="wikitable"
{| class="wikitable"
|-
|-
| | scale name
! | scale name
| | sL pattern
! | sL pattern
| | example in C
! | example in C
| | 1st chain
! | 1st chain
| | 2nd chain
! | 2nd chain
| | 3rd chain
! | 3rd chain
| | 4th chain
! | 4th chain
|-
|-
| | 1st Diminished[ 8]
| | 1st Diminished[ 8]
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|}
|}


There are only two [[Blackwood|Blackwood]] [10] modes. The period is a fifth-octave = 240¢. The generator is a just 5/4 = 386¢. L = 146¢ and s = 94¢. The lattice can be expressed using a 3\5 period Using ups and downs as before with each genchain at a different "height":
There are only two [[Blackwood]] [10] modes. The period is a fifth-octave = 240¢. The generator is a just 5/4 = 386¢. L = 146¢ and s = 94¢. The lattice can be expressed using a 3\5 period Using ups and downs as before with each genchain at a different "height":


E^^ ------- G#^^
E^^ ------- G#^^
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{| class="wikitable"
{| class="wikitable"
|-
|-
| | scale name
! | scale name
| | sL pattern
! | sL pattern
| | example in C
! | example in C
| | genchains
! | genchains
|-
|-
| | 1st Blackwood [10]
| | 1st Blackwood [10]
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|}
|}


='''<u>Other rank-2 scales</u>'''=
=Other rank-2 scales=


These are scales that are neither MOS nor MODMOS. Some scales have too many or too few notes. If they have an unbroken genchain, they can be named Meantone [6], Meantone [8], etc.
These are scales that are neither MOS nor MODMOS. Some scales have too many or too few notes. If they have an unbroken genchain, they can be named Meantone [6], Meantone [8], etc.
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{| class="wikitable"
{| class="wikitable"
|-
|-
| | scale
! | scale
| | genchain
! | genchain
| | name
! | name
| | sMLX pattern
! | sMLX pattern
|-
|-
| | octotonic:
| | '''octotonic:'''
| |  
| |  
| |  
| |  
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| | LMLs MXM
| | LMLs MXM
|-
|-
| | nonotonic:
| | '''nonatonic:'''
| |  
| |  
| |  
| |  
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| | LMLsM MLsM
| | LMLsM MLsM
|-
|-
| | hexatonic:
| | '''hexatonic:'''
| |  
| |  
| |  
| |  
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| | MLM MMs
| | MLM MMs
|-
|-
| | pentatonic:
| | '''pentatonic:'''
| |  
| |  
| |  
| |  
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|}
|}


Even 7-note scales can be non-MOS and non-MODMOS. For example, A C D D# E F G# A. The genchain is F C * D A E * * * G# D#. The name requires alterations, adds and drops: A 5th Meantone [7] #7 no2 add #4.
Even 7-note scales can be non-MOS and non-MODMOS. For example, A C D D# E F G# A. The genchain is F C * D A E * * * G# D#. The name requires alterations, adds and drops: A 5th Meantone[7] #7 no2 add #4.


Another possibility is a scale that would be MOS, but the generator is too sharp or flat. For example, a genchain F C G D A E B of 8\13 fifths makes an out-of-order scale A C B D F E G A. This scale is best named as Meantone [5] with added notes: Which brings us to...
Another possibility is a scale that would be MOS, but the generator is too sharp or flat. For example, a genchain F C G D A E B of 8\13 fifths makes an out-of-order scale A C B D F E G A. This scale is best named as Meantone [5] with added notes: Which brings us to...


=<u>Non-heptatonic Scales</u>=
=Non-heptatonic Scales=


As long as we stick to MOS scales, terms like Meantone [5] or Meantone {6} are fine. But when we alter, add or drop notes, we need to define what something like "#5" means in a pentatonic or hexatonic context.
As long as we stick to MOS scales, terms like Meantone [5] or Meantone {6} are fine. But when we alter, add or drop notes, we need to define what something like "#5" means in a pentatonic or hexatonic context.
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{| class="wikitable"
{| class="wikitable"
|-
|-
| | notation
! | notation
| | scale name
! | scale name
| | sL pattern
! | sL pattern
| | example in C
! | example in C
| | genchain
! | genchain
|-
|-
| | heptatonic
| | heptatonic
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The heptatonic-notated MODMOS has "+7" because B is the 7th letter from C. Likewise octotonic has "+8" because with H, B is the 8th letter.
The heptatonic-notated MODMOS has "+7" because B is the 7th letter from C. Likewise octotonic has "+8" because with H, B is the 8th letter.


=<u>Rationale</u>=
=Rationale=


'''<u>Why not number the modes in the order they occur in the scale?</u>'''
'''Why not number the modes in the order they occur in the scale?'''


Scale-based numbering would order the modes Ionian, Dorian, Phrygian, etc.
Scale-based numbering would order the modes Ionian, Dorian, Phrygian, etc.
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The disadvantage of genchain-based numbering is that the mode numbers are harder to relate to the scale. However this is arguably an advantage, because in the course of learning to relate the mode numbers, one internalizes the genchain.
The disadvantage of genchain-based numbering is that the mode numbers are harder to relate to the scale. However this is arguably an advantage, because in the course of learning to relate the mode numbers, one internalizes the genchain.


<u>'''Why make an exception for 3/2 vs 4/3 as the generator?'''</u>
'''Why make an exception for 3/2 vs 4/3 as the generator?'''
 
There are centuries of established thought that the fifth, not the fourth, generates the pythagorean, meantone and well tempered scales, as these quotes show (emphasis added):


"Pythagorean tuning is a tuning of the syntonic temperament in which the <span style="">generator</span> is the ratio <u>'''<span style="">3:2</span>'''</u> (i.e., the untempered perfect <u>'''fifth'''</u>)." -- [https://en.wikipedia.org/wiki/Pythagorean_tuning en.wikipedia.org/wiki/Pythagorean_tuning]
There are centuries of established thought that the fifth, not the fourth, generates the Pythagorean, meantone and well tempered scales, as these quotes show (emphasis added):


"The syntonic temperament is a system of musical tuning in which the frequency ratio of each musical interval is a product of powers of an octave and a tempered perfect <u>'''fifth'''</u>." -- [https://en.wikipedia.org/wiki/Syntonic_temperament en.wikipedia.org/wiki/Syntonic_temperament]
"Pythagorean tuning is a tuning of the syntonic temperament in which the generator is the ratio '''3:2''' (i.e., the untempered perfect '''fifth''')." -- [https://en.wikipedia.org/wiki/Pythagorean_tuning]


"Meantone is constructed the same way as Pythagorean tuning, as a stack of perfect <u>'''fifths'''</u>." --
"The syntonic temperament is a system of musical tuning in which the frequency ratio of each musical interval is a product of powers of an octave and a tempered perfect '''fifth'''." -- [https://en.wikipedia.org/wiki/Syntonic_temperament]


[https://en.wikipedia.org/wiki/Meantone_temperament en.wikipedia.org/wiki/Meantone_temperament]
"Meantone is constructed the same way as Pythagorean tuning, as a stack of perfect '''fifths'''." -- [https://en.wikipedia.org/wiki/Meantone_temperament]


"In this system the perfect <u>'''fifth'''</u> is flattened by one quarter of a syntonic comma." -- [https://en.wikipedia.org/wiki/Quarter-comma_meantone en.wikipedia.org/wiki/Quarter-comma_meantone]
"In this system the perfect '''fifth''' is flattened by one quarter of a syntonic comma." -- [https://en.wikipedia.org/wiki/Quarter-comma_meantone]


"The term "well temperament" or "good temperament" usually means some sort of <span style="">irregular temperament</span> in which the tempered <u>'''fifths'''</u> are of different sizes." -- [https://en.wikipedia.org/wiki/Well_temperament en.wikipedia.org/wiki/Well_temperament]
"The term "well temperament" or "good temperament" usually means some sort of irregular temperament in which the tempered '''fifths''' are of different sizes." -- [https://en.wikipedia.org/wiki/Well_temperament]


"A foolish consistency is the hobgoblin of little minds". To choose 4/3 over 3/2 merely for the sake of consistency would be pointless. Unlike a <u>wise</u> consistency, it wouldn't reduce memorization, because it's well known that the generator is historically 3/2.
"A foolish consistency is the hobgoblin of little minds". To choose 4/3 over 3/2 merely for the sake of consistency would be pointless. Unlike a <u>wise</u> consistency, it wouldn't reduce memorization, because it's well known that the generator is historically 3/2.


<u>'''Then why not always choose the larger of the two generators?'''</u>
'''Then why not always choose the larger of the two generators?'''


Interval arithmetic is easier with smaller intervals. It's easier to add up stacked 2nds than stacked 7ths. Also, when the generator is a 2nd, the genchain is often identical to the scale, simplifying mode numbering. (See Porcupine [7] above.)
Interval arithmetic is easier with smaller intervals. It's easier to add up stacked 2nds than stacked 7ths. Also, when the generator is a 2nd, the genchain is often identical to the scale, simplifying mode numbering. (See Porcupine [7] above.)


'''<u>Why not always choose the chroma-positive generator?</u>'''
'''Why not always choose the chroma-positive generator?'''


See below.
See below.


<u>'''Why not just use UDP notation?'''</u>
'''Why not just use UDP notation?'''


One problem with [[Modal_UDP_Notation|UDP]] is that avoiding chroma-negative generators causes the genchain to reverse direction frequently as you lengthen or shorten it, which affects the mode names. If exploring the various MOS's of a temperament, one has to constantly check the genchain direction. In Mode Numbers notation, the direction is unchanging.
One problem with [[Modal_UDP_Notation|UDP]] is that avoiding chroma-negative generators causes the genchain to reverse direction frequently as you lengthen or shorten it, which affects the mode names. If exploring the various MOS's of a temperament, one has to constantly check the genchain direction. In Mode Numbers notation, the direction is unchanging.
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{| class="wikitable"
{| class="wikitable"
|-
|-
| | scale
! | scale
| | UDP generator
! | UDP generator
| | UDP genchain
! | UDP genchain
| | Mode Numbers generator
! | Mode Numbers generator
| | Mode Numbers genchain
! | Mode Numbers genchain
|-
|-
| | Meantone[5] in 31edo
| | Meantone[5] in 31edo
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{| class="wikitable"
{| class="wikitable"
|-
|-
| | scale
! | scale
| | UDP genchain
! | UDP genchain
| | Mode Numbers genchain
! | Mode Numbers genchain
|-
|-
| | Meantone [2]
| | Meantone [2]
Line 995: Line 992:
Furthermore, UDP uses the more mathematical [https://en.wikipedia.org/wiki/Zero-based_numbering zero-based counting] and Mode Numbers notation uses the more intuitive one-based counting. UDP is mathematician-oriented whereas Mode Numbers notation is musician-oriented.
Furthermore, UDP uses the more mathematical [https://en.wikipedia.org/wiki/Zero-based_numbering zero-based counting] and Mode Numbers notation uses the more intuitive one-based counting. UDP is mathematician-oriented whereas Mode Numbers notation is musician-oriented.


'''<u>Related links</u>:'''
=Related links=


Jake Freivald has his own method of naming modes here:
Jake Freivald has his own method of naming modes here:


[[Naming_Rank-2_Scales#Jake Freivald method|http://xenharmonic.wikispaces.com/Naming+Rank-2+Scales#Jake%20Freivald%20method ]]
[[Naming_Rank-2_Scales#Jake Freivald method|http://xenharmonic.wikispaces.com/Naming+Rank-2+Scales#Jake%20Freivald%20method]]


[[Category:Rank-2]]
[[Category:Rank 2]]
[[category:naming]]
Retrieved from "https://en.xen.wiki/w/Kite%27s_Genchain_mode_numbering"