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Moved section on how to find mos intervals to appendix page; algorithms may need to be verified to show that they work as intended
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{{Main|TAMNAMS}}
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This page is an appendix to '''TAMNAMS'''.
This page is an appendix to '''TAMNAMS'''.


== Reasoning for step ratio names ==
== Reasoning for step ratio names ==
 
=== Derivation of the step ratio names ===
===Derivation of the step ratio names===
The idea is to start with the simplest ratios ({{nowrap|L/s {{=}} 1/0}} and {{nowrap|L/s {{=}} 1/1}}) and derive more complex ratios through repeated application of the [[mediant]] (aka Farey addition) to adjacent fractions.
The idea is to start with the simplest ratios (L/s = 1/0 and L/s = 1/1) and derive more complex ratios through repeated application of the [[mediant]] (aka Farey addition) to adjacent fractions.
* Applying the mediant to the starting intervals 1/0 and 1/1 gives {{nowrap|(1 + 1)/(1 + 0) {{=}} 2/1}}, and as this is the simplest possible ratio where the large and small step are distinguished and nonzero, it is called the ''basic'' tuning. (Note that if applying the mediant to 1/0 seems confusing, think of it as equivalent to applying the mediant to 0/1 and 1/1 and the ratios as flipped, thus representing s/L rather than L/s when written this way.)
*Applying the mediant to the starting intervals 1/0 and 1/1 gives (1+1)/(1+0) = 2/1, and as this is the simplest possible ratio where the large and small step are distinguished and nonzero, it is called the ''basic'' tuning. (Note that if applying the mediant to 1/0 seems confusing, think of it as equivalent to applying the mediant to 0/1 and 1/1 and the ratios as flipped, thus representing s/L rather than L/s when written this way.)
* As {{nowrap|L/s {{=}} 1/1}} represents L and s being equal in size, it is called ''equalized''.
 
* As {{nowrap|L/s {{=}} 1/0}} represents {{nowrap|s {{=}} 0}}, it is called ''collapsed'', as the small scale steps collapse to zero cents and disappear.
*As L/s = 1/1 represents L and s being equal in size, it is called ''equalized''.
* The mediant of 1/1 and 2/1 is 3/2, thus making the scale sound mellower/softer, and as this is the simplest (in the sense of lowest [[Odd limit#Relationship_to_other_limits|integer limit]]) ratio to represent such a property, it is simply called the ''soft'' tuning.
 
* Analogously, the mediant of 2/1 and 1/0, 3/1, is called the ''hard'' tuning. Thus you can say that a step ratio tuning is ''hard of'' or ''soft of'' another step ratio tuning.
*As L/s = 1/0 represents s = 0, it is called ''collapsed'', as the small scale steps collapse to zero cents and disappear.
* To get something between soft and basic we take the mediant again and get 5/3 for ''semisoft'', and analogously 5/2 for ''semihard''. To get something more extreme we take the mediant of 1/0 with 3/1 for a harder-than-hard tuning, giving us 4/1 for ''superhard'' and analogously 4/3 for ''supersoft''.
 
*The mediant of 1/1 and 2/1 is 3/2, thus making the scale sound mellower/softer, and as this is the simplest (in the sense of lowest [[Odd limit#Relationship_to_other_limits|integer limit]]) ratio to represent such a property, it is simply called the ''soft'' tuning.
 
*Analogously, the mediant of 2/1 and 1/0, 3/1, is called the ''hard'' tuning. Thus you can say that a step ratio tuning is ''hard of'' or ''soft of'' another step ratio tuning.
 
*To get something between soft and basic we take the mediant again and get 5/3 for ''semisoft'', and analogously 5/2 for ''semihard''. To get something more extreme we take the mediant of 1/0 with 3/1 for a harder-than-hard tuning, giving us 4/1 for ''superhard'' and analogously 4/3 for ''supersoft''.
There are also tertiary names beyond the above:
There are also tertiary names beyond the above:
*Anything softer than supersoft is ''ultrasoft,'' and anything harder than superhard is ''ultrahard''. Something between soft and supersoft is ''parasoft'', as ''para-'' means both ''beyond'' and ''next to''. Something between hard and superhard is ''parahard''.
* Anything softer than supersoft is ''ultrasoft,'' and anything harder than superhard is ''ultrahard''. Something between soft and supersoft is ''parasoft'', as ''para-'' means both ''beyond'' and ''next to''. Something between hard and superhard is ''parahard''.
* Something between soft and basic is ''hyposoft'' as it is less soft than soft. Something between hard and basic is ''hypohard'' for the same reason. Between semisoft and basic is ''minisoft'' and between semihard and basic is ''minihard''.
* Finally, between soft and semisoft is ''quasisoft'' as such scales may potentially be mistaken for soft or semisoft while not being either—hence the use of the prefix ''quasi-'', and between hard and semihard is ''quasihard'' for the same reason.
The reasoning for the ''para-/super-/ultra-'' progression (note that ''super-'' is the odd one out as it refers to an exact ratio) is it mirrors naming for shades of musical intervals and because ''[[parapythagorean]]'' is between ''pythagorean'' and ''[[superpythagorean]]''.


*Something between soft and basic is ''hyposoft'' as it is less soft than soft. Something between hard and basic is ''hypohard'' for the same reason. Between semisoft and basic is ''minisoft'' and between semihard and basic is ''minihard''.
This results in the ''central spectrum''—an elegant system which names all exact L/s ratios in the 5-integer-limit excepting only 5/1 and 5/4 which are disincluded intentionally for a variety of reasons: to keep the maximum corresponding notes per period in an [[EPD|equal pitch division]] low, because it keeps the 'tree' of mediants complete to a certain number of layers, and because their disinclusion gives a roughly-equally-spaced set of ratios, with the regions between 4/3 and 1/1 and between 4/1 and 1/0 being the only exceptions—corresponding to extreme tunings. Note that filling in those extreme regions is the purpose of the extended spectrum


*Finally, between soft and semisoft is ''quasisoft'' as such scales may potentially be mistaken for soft or semisoft while not being either - hence the use of the prefix ''quasi-'', and between hard and semihard is ''quasihard'' for the same reason.
=== Extending the spectrum's edges ===
The reasoning for the ''para- super- ultra-'' progression (note that ''super-'' is the odd one out as it refers to an exact ratio) is it mirrors naming for shades of musical intervals and because ''parapythagorean'' is between ''pythagorean'' and ''superpythagorean''.
 
This results in the ''central spectrum'' - an elegant system which names all exact L/s ratios in the 5-integer-limit excepting only 5/1 and 5/4 which are disincluded intentionally for a variety of reasons: to keep the maximum corresponding notes per period in an [[EPD|equal pitch division]] low, because it keeps the 'tree' of mediants complete to a certain number of layers, and because their disinclusion gives a roughly-equally-spaced set of ratios, with the regions between 4/3 and 1/1 and between 4/1 and 1/0 being the only exceptions - corresponding to extreme tunings. Note that filling in those extreme regions is the purpose of the extended spectrum.
===Extending the spectrum's edges===
Extending the spectrum builds on the central spectrum and relies on a few key observations.
Extending the spectrum builds on the central spectrum and relies on a few key observations.


Firstly, as periods and mosses come in wildly different shapes and sizes, and as we want to represent a somewhat representative variety of ''simple'' tunings for the step ratio for a given mos pattern and period, the notion of ''simple'' used will correspond to the number of equally-spaced tones per period required. This is expressed as [number of large steps in pattern]*L + [number of small steps in pattern]*s, where L and s are from the step ratio itself, L/s, and are assumed to be coprime. Then, in order to not introduce bias to mos patterns with more L's or more s's, we should assume that both are equally likely and thus weight both equally, which means that the resulting minimum number of tones per period for a ratio L/s is L+s.
Firstly, as periods and mosses come in wildly different shapes and sizes, and as we want to represent a somewhat representative variety of ''simple'' tunings for the step ratio for a given mos pattern and period, the notion of ''simple'' used will correspond to the number of equally-spaced tones per period required. This is expressed as {{nowrap|''x''L + ''y''s}}, where ''x'' and ''y'' are the number of large and small steps in the scale, and where L and s are from the step ratio L/s (where L and s are assumed to be coprime). Then, in order to not introduce bias to mos patterns with more L's or more s's, we should assume that both are equally likely and thus weight both equally, which means that the resulting minimum number of tones per period for a ratio L/s is {{nowrap|L + s}}.


The next observation is that the large values of L/s can be a lot more consequential than the ones close to 1/1 due to the fact that small steps are guaranteed to be smaller than large steps and that we don't know how many small steps there are compared to large steps, and therefore the ''hard'' end of the spectrum is more vast, and analogously, L/s values close to 1/1 will tend to be inconsequential and for very close values likely impractical to distinguish - in the extremes only serving small tuning adjustments rather than melodic properties. This leads to another observation: mos patterns with periods tuned to step ratios, while related to temperaments, ''are not'' temperaments - instead forming a sort of amalgamative superset of temperaments if you want to force a temperament interpretation, and thus their main function is in melodic structure, with temperaments informing potential harmonies and microtunings. Thus, the spectrum should be kept minimal and simple so that it is both generally hearable and not too specific.
The next observation is that the large values of L/s can be a lot more consequential than the ones close to 1/1 due to the fact that small steps are guaranteed to be smaller than large steps and that we don't know how many small steps there are compared to large steps, and therefore the ''hard'' end of the spectrum is more vast, and analogously, L/s values close to 1/1 will tend to be inconsequential and for very close values likely impractical to distinguish—in the extremes only serving small tuning adjustments rather than melodic properties. This leads to another observation: mos patterns with periods tuned to step ratios, while related to temperaments, ''are not'' temperaments—instead forming a sort of amalgamative superset of temperaments if you want to force a temperament interpretation, and thus their main function is in melodic structure, with temperaments informing potential harmonies and microtunings. Thus, the spectrum should be kept minimal and simple so that it is both generally hearable and not too specific.


The most obvious adjustment to the edges is to draw a distinction between ''ultrasoft'' and ''pseudoequalized'' by adding a step ratio corresponding to ''semiequalized'', and between ''ultrahard'' and ''pseudocollapsed'' by adding a step ratio corresponding to ''semicollapsed''. Thus:
The most obvious adjustment to the edges is to draw a distinction between ''ultrasoft'' and ''pseudoequalized'' by adding a step ratio corresponding to ''semiequalized'', and between ''ultrahard'' and ''pseudocollapsed'' by adding a step ratio corresponding to ''semicollapsed''. Thus:
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'''Ultrahard''' is between '''superhard''' and '''semicollapsed''', and '''pseudocollapsed''' is between '''semicollapsed''' and '''collapsed'''.
'''Ultrahard''' is between '''superhard''' and '''semicollapsed''', and '''pseudocollapsed''' is between '''semicollapsed''' and '''collapsed'''.


Then all that's left is to decide what the step ratios for semicollapsed and semiequalized should be. In order to keep the spacing (of the s/L ratios when graphed, or to a lesser extent the L/s ratios if you see the roughly gradual increase in spacing in that form) roughly consistent with all the other ratios, '''semiequalized''' should be L/s = 6/5 rather than L/s = 5/4. Then note the complexity of L/s = 6/5 is 6+5=11, so to find the corresponding complexity for '''semicollapsed''' we use L/s = 10/1 as 10+1=11 too. Then finally, to preserve some of the symmetry, we include L/s = 6/1 as '''extrahard'''. Although L/s = 10/1 for '''semicollapsed''' may seem a little extreme of a boundary, L/s = 12/1 would actually be what is the most ''equally spaced'' continuing on from 6/1 for the same reason that L/s = 6/5 is the most ''equally spaced''. Note that while the range from '''superhard''' to '''semicollapsed''' is '''ultrahard''', the region may be split into two sub-ranges:
Then all that's left is to decide what the step ratios for semicollapsed and semiequalized should be. In order to keep the spacing (of the s/L ratios when graphed, or to a lesser extent the L/s ratios if you see the roughly gradual increase in spacing in that form) roughly consistent with all the other ratios, '''semiequalized''' should be {{nowrap|L/s {{=}} 6/5}} rather than {{nowrap|L/s {{=}} 5/4}}. Then note the complexity of {{nowrap|L/s {{=}} 6/5}} is {{nowrap|6 + 5 {{=}} 11}}, so to find the corresponding complexity for '''semicollapsed''' we use {{nowrap|L/s {{=}} 10/1}} as {{nowrap|10 + 1 {{=}} 11}} too. Then finally, to preserve some of the symmetry, we include {{nowrap|L/s {{=}} 6/1}} as '''extrahard'''. Although {{nowrap|L/s {{=}} 10/1}} for '''semicollapsed''' may seem a little extreme of a boundary, {{nowrap|L/s {{=}} 12/1}} would actually be what is the most ''equally spaced'' continuing on from 6/1 for the same reason that {{nowrap|L/s {{=}} 6/5}} is the most ''equally spaced''. Note that while the range from '''superhard''' to '''semicollapsed''' is '''ultrahard''', the region may be split into two sub-ranges:


'''superhard''' (L/s=4/1) to '''extrahard''' (L/s=6/1) is '''hyperhard''' (4 < L/s < 6).
'''superhard''' ({{nowrap|L/s {{=}} 4/1}}) to '''extrahard''' ({{nowrap|L/s {{=}} 6/1}}) is '''hyperhard''' ({{nowrap|4 &lt; L/s &lt; 6}}).


'''extrahard''' (L/s=6/1) to '''semicollapsed''' (L/s=10/1) is '''clustered''' (6 < L/s < 10).
'''extrahard''' ({{nowrap|L/s {{=}} 6/1}}) to '''semicollapsed''' ({{nowrap|L/s {{=}} 10/1}}) is '''clustered''' ({{nowrap|6 &lt; L/s &lt; 10}}).


With the inclusion of these 3 new L/s ratios nearer the edges of the spectrum and names for the range divisions they create, we get the extended spectrum, summarised and detailed above, just for the regions affected to avoid repetition.
With the inclusion of these 3 new L/s ratios nearer the edges of the spectrum and names for the range divisions they create, we get the extended spectrum, summarised and detailed above, just for the regions affected to avoid repetition.
===Extended spectrum===
 
{| class="wikitable"
=== Extended spectrum ===
|+Extended spectrum of step ratio ranges and specific step ratios
{| class="wikitable" style="text-align: center;"
|+ style="font-size: 105%;" | Extended spectrum of step ratio ranges and specific step ratios
|-
|-
! colspan="3" |Central ranges
! colspan="3" | Central ranges
! colspan="2" |Extended ranges
! colspan="2" | Extended ranges
!Specific step ratios
! Specific<br />step ratios
!Notes
! Notes
|-
|-
|
|
|
|
|
|
| colspan="2" |
| colspan="2" |  
|'''1:1 (equalized)'''
| '''1:1<br />(equalized)'''
|Trivial/pathological
| Trivial/pathological
|-
|-
| rowspan="9" |1:1 to 2:1 (soft-of-basic)
| rowspan="9" | 1:1 to 2:1<br />(soft-of-basic)
| colspan="2" rowspan="3" |1:1 to 4:3 (ultrasoft)
| colspan="2" rowspan="3" | 1:1 to 4:3<br />(ultrasoft)
| colspan="2" |1:1 to 6:5 (pseudoequalized)
| colspan="2" | 1:1 to 6:5<br />(pseudoequalized)
|
|
|
|
|-
|-
| colspan="2" |
| colspan="2" |  
|'''6:5 (semiequalized)'''
| '''6:5<br />(semiequalized)'''
|
|  
|-
|-
| colspan="2" |6:5 to 4:3 (ultrasoft)
| colspan="2" | 6:5 to 4:3<br />(ultrasoft)
|
|  
|
|  
|-
|-
|
|  
|
|  
| colspan="2" |
| colspan="2" |  
|'''4:3 (supersoft)'''
| '''4:3<br />(supersoft)'''
| rowspan="13" |Nonextreme range, as detailed by central spectrum
| rowspan="13" | Nonextreme range, as detailed<br />by central spectrum
|-
|-
| colspan="2" |4:3 to 3:2 (parasoft)
| colspan="2" | 4:3 to 3:2<br />(parasoft)
| colspan="2" |4:3 to 3:2 (parasoft)
| colspan="2" | 4:3 to 3:2<br />(parasoft)
|
|  
|-
|-
|
|  
|
|  
| colspan="2" |
| colspan="2" |  
|'''3:2 (soft)'''
| '''3:2<br />(soft)'''
|-
|-
| rowspan="3" |3:2 to 2:1 (hyposoft)
| rowspan="3" | 3:2 to 2:1<br />(hyposoft)
|3:2 to 5:3 (quasisoft)
| 3:2 to 5:3<br />(quasisoft)
| colspan="2" |3:2 to 5:3 (quasisoft)
| colspan="2" | 3:2 to 5:3<br />(quasisoft)
|
|  
|-
|-
|
|  
| colspan="2" |
| colspan="2" |  
|'''5:3 (semisoft)'''
| '''5:3<br />(semisoft)'''
|-
|-
|5:3 to 2:1 (minisoft)
| 5:3 to 2:1<br />(minisoft)
| colspan="2" |5:3 to 2:1 (minisoft)
| colspan="2" | 5:3 to 2:1<br />(minisoft)
|
|  
|-
|-
|
|  
|
|  
|
|  
| colspan="2" |
| colspan="2" |  
|'''2:1 (basic)'''
| '''2:1<br />(basic)'''
|-
|-
| rowspan="11" |2:1 to 1:0 (hard-of-basic)
| rowspan="11" | 2:1 to 1:0<br />(hard-of-basic)
| rowspan="3" |2:1 to 3:1 (hypohard)
| rowspan="3" | 2:1 to 3:1<br />(hypohard)
|2:1 to 5:2 (minihard)
| 2:1 to 5:2<br />(minihard)
| colspan="2" |2:1 to 5:2 (minihard)
| colspan="2" | 2:1 to 5:2<br />(minihard)
|
|  
|-
|-
|
|  
| colspan="2" |
| colspan="2" |  
|'''5:2 (semihard)'''
| '''5:2<br />(semihard)'''
|-
|-
|5:2 to 3:1 (quasihard)
| 5:2 to 3:1<br />(quasihard)
| colspan="2" |5:2 to 3:1 (quasihard)
| colspan="2" | 5:2 to 3:1<br />(quasihard)
|
|  
|-
|-
|
|  
|
|  
| colspan="2" |
| colspan="2" |  
|'''3:1 (hard)'''
| '''3:1<br />(hard)'''
|-
|-
| colspan="2" |3:1 to 4:1 (parahard)
| colspan="2" | 3:1 to 4:1<br />(parahard)
| colspan="2" |3:1 to 4:1 (parahard)
| colspan="2" | 3:1 to 4:1<br />(parahard)
|
|  
|-
|-
|
|  
|
|  
| colspan="2" |
| colspan="2" |  
|'''4:1 (superhard)'''
| '''4:1<br />(superhard)'''
|-
|-
| colspan="2" rowspan="5" |4:1 to 1:0 (ultrahard)
| colspan="2" rowspan="5" | 4:1 to 1:0<br />(ultrahard)
| rowspan="3" |4:1 to 10:1 (ultrahard)
| rowspan="3" | 4:1 to 10:1<br />(ultrahard)
|4:1 to 6:1 (hyperhard)
| 4:1 to 6:1<br />(hyperhard)
|
|  
|
|  
|-
|-
|
|  
|'''6:1 (extrahard)'''
| '''6:1<br />(extrahard)'''
|
|  
|-
|-
|6:1 to 10:1 (clustered)
| 6:1 to 10:1<br />(clustered)
|
|  
|
|  
|-
|-
| colspan="2" |
| colspan="2" |  
|'''10:1 (semicollapsed)'''
| '''10:1<br />(semicollapsed)'''
|
|  
|-
|-
| colspan="2" |10:1 to 1:0 (pseudocollapsed)
| colspan="2" | 10:1 to 1:0<br />(pseudocollapsed)
|
|  
|
|  
|-
|-
|
|  
|
|  
|
|  
| colspan="2" |
| colspan="2" |  
|'''1:0 (collapsed)'''
| '''1:0<br />(collapsed)'''
|Trivial/pathological
| Trivial/pathological
|}
|}
===Terminology and final notes===
A ratio of L/s = k/1 can be called ''k-hard'' and a ratio of L/s = k/(k-1) can analogously be called ''k-soft'', so the simplest ultrasoft tuning is 5-soft or ''pentasoft'', the simplest hyperhard tuning is 5-hard or ''pentahard'', the simplest clustered tuning is 7-hard or ''heptahard'', 8-hard is ''octahard'', 9-hard is ''nonahard'', and finally, the characteristic simple ultrahard tuning is 6-hard or ''extrahard'', as previously discussed, which can be seen to be similar to ''hexahard'' - hopefully helping with memorisation.


A perhaps useful (or otherwise mildly amusing) mnemonic is ''2-soft is too soft to be hard and 2-hard is too hard to be soft'', representing that 2-soft = 2-hard = 2/1 = '''basic'''.
=== Terminology and final notes ===
A ratio of {{nowrap|L/s {{=}} ''k''/1}} can be called ''k-hard'' and a ratio of {{nowrap|L/s {{=}} ''k''/(''k'' − 1)}} can analogously be called ''k-soft'', so the simplest ultrasoft tuning is 5-soft or ''pentasoft'', the simplest hyperhard tuning is 5-hard or ''pentahard'', the simplest clustered tuning is 7-hard or ''heptahard'', 8-hard is ''octahard'', 9-hard is ''nonahard'', and finally, the characteristic simple ultrahard tuning is 6-hard or ''extrahard'', as previously discussed, which can be seen to be similar to ''hexahard''&mdash;hopefully helping with memorisation.


Note that often the central spectrum will be sufficient for exploring a mos pattern-period combination, and the extended spectrum is intended more for (literally) edge cases where it may be useful. Often if a temperament interpretation doesn't seem to show up for a mos  pattern-period combination, it just means the temperament needs a more complex mos pattern to narrow down the generator range. An example of this phenomena is the highly complex mos pattern of [[12L 17s]] represents near-Pythagorean tunings well due to having a generator of a fourth or a fifth bounded between those of [[12edo]] and those of [[29edo]], which are roughly equally off but in opposite directions, and many important near-Pythagorean systems show up in just the ratios of the central spectrum alone.
A perhaps useful (or otherwise mildly amusing) mnemonic is ''2-soft is too soft to be hard and 2-hard is too hard to be soft'', representing that {{nowrap|2-soft {{=}} 2-hard {{=}} 2/1 {{=}} '''basic'''}}.
 
Note that often the central spectrum will be sufficient for exploring a mos pattern-period combination, and the extended spectrum is intended more for (literally) edge cases where it may be useful. Often if a temperament interpretation doesn't seem to show up for a mos  pattern-period combination, it just means the temperament needs a more complex mos pattern to narrow down the generator range. An example of this phenomena is the highly complex mos pattern of [[12L&nbsp;17s]] represents near-Pythagorean tunings well due to having a generator of a fourth or a fifth bounded between those of [[12edo]] and those of [[29edo]], which are roughly equally off but in opposite directions, and many important near-Pythagorean systems show up in just the ratios of the central spectrum alone.


== Reasoning for mos interval names ==
== Reasoning for mos interval names ==
 
=== Reasoning for 0-indexed intervals ===
===Reasoning for 0-indexed intervals===
Note that a unison is a 0-mosstep, rather than a mos1st; likewise, the term 1-mosstep is used rather than a mos2nd. One might be tempted to generalize diatonic 1-indexed ordinal names: ''In 31edo's ultrasoft [[mosh]] scale, the perfect mosthird (aka Pmosh3rd) is a neutral third and the major mosfifth (aka Lmosh5th) is a perfect fifth.'' The way intervals are named above (and in 12edo theory) has a problem. An interval that's n steps wide is named ''(n+1)th''. This means that adding two intervals is more complicated than it should be. Stacking two fifths makes a ninth, when naively it would make a tenth. We're used to this for the diatonic scale, but when dealing with unfamiliar scale structures, it can be very confusing.
Note that a unison is a 0-mosstep, rather than a mos1st; likewise, the term 1-mosstep is used rather than a mos2nd. One might be tempted to generalize diatonic 1-indexed ordinal names: ''In 31edo's ultrasoft [[mosh]] scale, the perfect mosthird (aka Pmosh3rd) is a neutral third and the major mosfifth (aka Lmosh5th) is a perfect fifth.'' The way intervals are named above (and in 12edo theory) has a problem. An interval that's n steps wide is named ''(n+1)th''. This means that adding two intervals is more complicated than it should be. Stacking two fifths makes a ninth, when naively it would make a tenth. We're used to this for the diatonic scale, but when dealing with unfamiliar scale structures, it can be very confusing.


To overcome this, TAMNAMS uses a 0-indexed name system for non-diatonic mos intervals, which makes the arithmetic needed to understand mos intervals much smoother. Going up a 0-mosstep means to go up zero steps, and stacking two 4-mossteps produces an 8-mosstep, rather than stacking two mos5ths to produce a mos9th. The use of ordinal indexing is generally discouraged when referring to non-diatonic mos intervals.
To overcome this, TAMNAMS uses a 0-indexed name system for non-diatonic mos intervals, which makes the arithmetic needed to understand mos intervals much smoother. Going up a 0-mosstep means to go up zero steps, and stacking two 4-mossteps produces an 8-mosstep, rather than stacking two mos5ths to produce a mos9th. The use of ordinal indexing is generally discouraged when referring to non-diatonic mos intervals.


=== How to find mos interval sizes ===
=== Deriving the names for mossteps ===
To find what mos interval sizes are found in a mos, start with the patterns of large and small steps that represents the mos in its brightest mode and its darkest mode (which is the reverse pattern for the mos's brightest mode). The idea of [[Recursive structure of MOS scales|mos recursion]] may be of help with finding the generators of a mos. Likewise, the idea of modal brightness and [[Modal UDP notation|UDP]] may be of help for a mos's modes.
To find what mos interval sizes are found in a mos, start with the patterns of large and small steps that represents the mos in its brightest mode and its darkest mode (which is the reverse pattern for the mos's brightest mode). The idea of [[Recursive structure of MOS scales|mos recursion]] may be of help with finding the generators of a mos. Likewise, the idea of modal brightness and [[Modal UDP notation|UDP]] may be of help for a mos's modes.


Given the mos ''x''L ''y''s, the following algorithm is used to find the '''brightest mode''' for that mos.
Given the mos ''x''L&nbsp;''y''s, the following algorithm is used to find the '''brightest mode''' for that mos.


# If either ''x'' or ''y'' is equal to 1 (base cases):
# If either ''x'' or ''y'' is equal to 1 (base cases):
## If both ''x'' and ''y'' are equal to 1, then the final scale is "Ls".
## If both ''x'' and ''y'' are equal to 1, then the final scale is "Ls".
## If only ''x'' is equal to 1, then the final scale is L followed by ''y'' s's.
## If only ''x'' is equal to 1, then the final scale is L followed by ''y''&nbsp;s's.
## If only ''y'' is equal to 1, then the final scale is ''x'' L's followed by s.
## If only ''y'' is equal to 1, then the final scale is ''x''&nbsp;L's followed by s.
# If neither ''x'' nor ''y'' is equal to 1 (recursive cases):
# If neither ''x'' nor ''y'' is equal to 1 (recursive cases):
## Let ''k'' be the greatest common factor of ''x'' and ''y''.
## Let ''k'' be the greatest common factor of ''x'' and ''y''.
## If ''x'' and ''y'' share a common factor ''k'', where ''k'' is greater than 1, then recursively call this algorithm to find the scale for (''x''/''k'')L (''y''/''k'')s; the final scale will be (''x''/''k'')L (''y''/''k'')s duplicated ''k'' times.
## If ''x'' and ''y'' share a common factor ''k'', where ''k'' is greater than 1, then recursively call this algorithm to find the scale for (''x''/''k'')L&nbsp;(''y''/''k'')s; the final scale will be (''x''/''k'')L&nbsp;(''y''/''k'')s duplicated ''k'' times.
## If ''x'' and ''y'' don't share a common factor that is greater than 1 (if ''x'' and ''y'' are coprime), then:
## If ''x'' and ''y'' don't share a common factor that is greater than 1 (if ''x'' and ''y'' are coprime), then:
### Let ''m1'' = min(''x'', ''y'') and ''m2'' = max(''x'', ''y'').
### Let {{nowrap|''m''<sub>1</sub> {{=}} min(''x'', ''y'')}} and {{nowrap|''m''<sub>2</sub> {{=}} max(''x'', ''y'')}}.
### Let ''z'' = ''m2'' mod ''m1'' and ''w'' = ''m1'' - ''z''.
### Let {{nowrap|''z'' {{=}} ''m''<sub>2</sub> mod ''m''<sub>1</sub>}} and {{nowrap|''w'' {{=}} ''m''<sub>1</sub> − ''z''}}.
### Let ''prescale'' be the mos string for ''z''L ''w''s. Recursively call this algorithm to find the scale for ''z''L ''w''s; the final scale will be based on this.
### Let ''prescale'' be the mos string for ''z''L&nbsp;''w''s. Recursively call this algorithm to find the scale for ''z''L&nbsp;''w''s; the final scale will be based on this.
### If ''x'' < ''y'', reverse the order of characters in the prescale. This is only needed if there are more L's than s's in the final scale.
### If {{nowrap|''x'' &lt; ''y''}}, reverse the order of characters in the prescale. This is only needed if there are more L's than s's in the final scale.
### To produce the final scale, the L's and s's of the prescale must be replaced with substrings consisting of L's and s's. Let ''u'' = ceil(''m2''/''m1'') and ''v'' = floor(''m2''/''m1'').
### To produce the final scale, the L's and s's of the prescale must be replaced with substrings consisting of L's and s's. Let {{nowrap|''u'' {{=}} {{ceil|''m''<sub>2</sub>/''m''<sub>1</sub>}}}} and {{nowrap|''v'' {{=}} {{floor|''m''<sub>2</sub>/''m''<sub>1</sub>}}}}.<ref group="note" name="floorceiling">{{ceil|&nbsp;}} denotes the ceiling function and {{floor|&nbsp;}} denotes the floor function.</ref>
#### If ''x'' > ''y'', every instance of an L in ''prescale'' is replaced with one L and ''u'' s's, and every s replaced with one L and ''v'' s's. This produces the final scale in its brightest mode.
#### If {{nowrap|''x'' &gt; ''y''}}, every instance of an L in ''prescale'' is replaced with one L and ''u''&nbsp;s's, and every s replaced with one L and ''v''&nbsp;s's. This produces the final scale in its brightest mode.
#### If ''y'' > ''x'', every instance of an L in ''prescale'' is replaced with ''u'' L's and one s, and every s replaced with ''v'' L's and one s. This produces the final scale in its brightest mode.
#### If {{nowrap|''x'' &lt; ''y''}}, every instance of an L in ''prescale'' is replaced with ''u''&nbsp;L's and one s, and every s replaced with ''v''&nbsp;L's and one s. This produces the final scale in its brightest mode.
 
Using 3L&nbsp;4s as an example, this is LsLsLss (brightest) and ssLsLsL (darkest). To find the large sizes of each ''k''-mosstep, consider the first ''k'' mossteps that make up the mos pattern for the brightest mode. Repeat this process with the mos pattern for the darkest mode to find each ''k''-mosstep's small size. To make these sizes more clear, we can denote the mos intervals as a sum of large and small steps {{nowrap|''i''L + ''j''s}}, where ''i'' and ''j'' are the number of L's and s's in the interval's step pattern; this is to say that the order of L's and s's doesn't matter, rather the amount of each step size. The large and small sizes should differ by replacing one L in the large size with an s.


Using 3L 4s as an example, this is LsLsLss (brightest) and ssLsLsL (darkest). To find the large sizes of each ''k''-mosstep, consider the first ''k'' mossteps that make up the mos pattern for the brightest mode. Repeat this process with the mos pattern for the darkest mode to find each ''k''-mosstep's small size. To make these sizes more clear, we can denote the mos intervals as a sum of large and small steps ''i''L+''j''s, where ''i'' and ''j'' are the number of L's and s's in the interval's step pattern; this is to say that the order of L's and s's doesn't matter, rather the amount of each step size. The large and small sizes should differ by replacing one L in the large size with an s.
{| class="wikitable"
{| class="wikitable"
! rowspan="2" |Interval
! colspan="2" |Large size (LsLsLss)
! colspan="2" |Small size (ssLsLsL)
|-
|-
!Step pattern
! rowspan="2" | Interval
!Sum
! colspan="2" | Large size (LsLsLss)
! colspan="2" | Small size (ssLsLsL)
|-
! Step pattern
! Sum
! Step pattern
! Step pattern
!Sum
! Sum
|-
|-
| 0-mosstep (unison)
| 0-mosstep (unison)
|none
| none
|'''0'''
| '''0'''
|none
| none
|'''0'''
| '''0'''
|-
|-
|1-mosstep
| 1-mosstep
|L
| L
|'''L'''
| '''L'''
|s
| s
|'''s'''
| '''s'''
|-
|-
| 2-mosstep
| 2-mosstep
|Ls
| Ls
|'''L+s'''
| '''L + s'''
|ss
| ss
|'''2s'''
| '''2s'''
|-
|-
|3-mosstep
| 3-mosstep
| LsL
| LsL
|'''2L+s'''
| '''2L + s'''
|ssL
| ssL
|'''1L+2s'''
| '''1L + 2s'''
|-
|-
| 4-mosstep
| 4-mosstep
|LsLs
| LsLs
|'''2L+2s'''
| '''2L + 2s'''
|ssLs
| ssLs
|'''1L+3s'''
| '''1L + 3s'''
|-
|-
|5-mosstep
| 5-mosstep
|LsLsL
| LsLsL
|'''3L+2s'''
| '''3L + 2s'''
|ssLsL
| ssLsL
|'''2L+3s'''
| '''2L + 3s'''
|-
|-
| 6-mosstep
| 6-mosstep
|LsLsLs
| LsLsLs
|'''3L+3s'''
| '''3L + 3s'''
|ssLsLs
| ssLsLs
|'''2L+4s'''
| '''2L + 4s'''
|-
|-
|7-mosstep (octave)
| 7-mosstep (octave)
|LsLsLss
| LsLsLss
|'''3L+4s'''
| '''3L + 4s'''
|ssLsLsL
| ssLsLsL
|'''3L+4s'''
| '''3L + 4s'''
|}
|}
Given the mos ''x''L ''y''s, the following algorithm is used to find the '''bright generator''' and its complement.
Given the mos ''x''L&nbsp;''y''s, the following algorithm is used to find the '''bright generator''' and its complement.


# If either ''x'' or ''y'' is equal to 1 (base cases):
# If either ''x'' or ''y'' is equal to 1 (base cases):
## If both ''x'' and ''y'' are equal to 1, then the generator is "L" and its complement is "s".
## If both ''x'' and ''y'' are equal to 1, then the generator is "L" and its complement is "s".
## If only ''x'' is equal to 1, then the generator is "L" followed by ''y-1'' s's, and the complement is "s".
## If only ''x'' is equal to 1, then the generator is "L" followed by {{nowrap|''y'' − 1}} s's, and the complement is "s".
## If only ''y'' is equal to 1, then the generator is "L" and the complement is ''x-1'' L's followed by "s".
## If only ''y'' is equal to 1, then the generator is "L" and the complement is {{nowrap|''x'' − 1}} L's followed by "s".
# If neither ''x'' nor ''y'' is equal to 1 (recursive cases):
# If neither ''x'' nor ''y'' is equal to 1 (recursive cases):
## Let ''k'' be the greatest common factor of ''x'' and ''y''.
## Let ''k'' be the greatest common factor of ''x'' and ''y''.
## If ''x'' and ''y'' share a common factor ''k'', where ''k'' is greater than 1, then recursively call this algorithm to find the generator and complement for (''x''/''k'')L (''y''/''k'')s; the intervals returned this way will apply to the period rather than the octave.
## If ''x'' and ''y'' share a common factor ''k'', where ''k'' is greater than 1, then recursively call this algorithm to find the generator and complement for {{nowrap|(''x''/''k'')L (''y''/''k'')s}}; the intervals returned this way will apply to the period rather than the octave.
## If ''x'' and ''y'' don't share a common factor that is greater than 1 (if ''x'' and ''y'' are coprime), then:
## If ''x'' and ''y'' don't share a common factor that is greater than 1 (if ''x'' and ''y'' are coprime), then:
### Let ''m1'' = min(''x'', ''y'') and ''m2'' = max(''x'', ''y'').
### Let {{nowrap|''m''<sub>1</sub> {{=}} min(''x'', ''y'')}} and {{nowrap|''m''<sub>2</sub> {{=}} max(''x'', ''y'')}}.
### Let ''z'' = ''m2'' mod ''m1'' and ''w'' = ''m1 - z''.
### Let {{nowrap|''z'' {{=}} ''m''<sub>2</sub> mod ''m''<sub>1</sub>}} and {{nowrap|''w'' {{=}} ''m1 − z''}}.
### Let ''gen'' be the scale's generator and ''comp'' be the generator's octave complement for the mos ''z''L ''w''s. Recursively call this algorithm to find these intervals for ''z''L ''w''s; the final scale's generator and complement will be based on this.
### Let ''gen'' be the scale's generator and ''comp'' be the generator's octave complement for the mos ''z''L&nbsp;''w''s. Recursively call this algorithm to find these intervals for ''z''L&nbsp;''w''s; the final scale's generator and complement will be based on this.
### If ''x'' < ''y'', reverse the order of characters in ''gen'' and ''comp'', then swap ''gen'' and ''comp''. This is only needed if there are more L's than s's in the scale.
### If {{nowrap|''x'' &lt; ''y''}}, reverse the order of characters in ''gen'' and ''comp'', then swap ''gen'' and ''comp''. This is only needed if there are more L's than s's in the scale.
### To produce the scale's generator and complement, the L's and s's of both intervals must be replaced with substrings consisting of L's and s's. Let ''u'' = ceil(''m2''/''m1'') and ''v'' = floor(''m2''/''m1'').
### To produce the scale's generator and complement, the L's and s's of both intervals must be replaced with substrings consisting of L's and s's. Let {{nowrap|''u'' {{=}} {{ceil|''m''<sub>2</sub>/''m''<sub>1</sub>}}}} and {{nowrap|''v'' {{=}} {{floor|''m''<sub>2</sub>/''m''<sub>1</sub>}}}}.<ref group="note" name="floorceiling" />
#### If ''x'' > ''y'', every instance of an L in both intervals is replaced with one L and ''u'' s's, and every s replaced with one L and ''v'' s's. This produces the final scale's generator and complement.
#### If {{nowrap|''x'' &gt; ''y''}}, every instance of an L in both intervals is replaced with one L and ''u'' s's, and every s replaced with one L and ''v'' s's. This produces the final scale's generator and complement.
#### If ''y'' > ''x'', every instance of an L in both intervals is replaced with ''u'' L's and one s, and every s replaced with ''v'' L's and one s. This produces the final scale's generator and complement.
#### If {{nowrap|''x'' &lt; ''y''}}, every instance of an L in both intervals is replaced with ''u'' L's and one s, and every s replaced with ''v'' L's and one s. This produces the final scale's generator and complement.
 
The length of ''gen'' is the number of mossteps for the bright generator, and the length of ''comp'' is the number of mossteps in the dark generator. For our example of 3L&nbsp;4s, the algorithm returns the step pattern '''Ls''' as the bright generator and '''LsLss''' as its complement, which are 2 and 5 mossteps wide, respectively. Since the large size of a bright generator is perfect and its small size diminished, and the large size of a dark generator is augmented and its small size perfect, the scale's generators can be identified as shown in the table.


The length of ''gen'' is the number of mossteps for the bright generator, and the length of ''comp'' is the number of mossteps in the dark generator. For our example of 3L 4s, the algorithm returns the step pattern '''Ls''' as the bright generator and '''LsLss''' as its complement, which are 2 and 5 mossteps wide, respectively. Since the large size of a bright generator is perfect and its small size diminished, and the small size of a dark generator is perfect and its small size perfect, the scale's generators can be identified as shown in the table.
{| class="wikitable"
{| class="wikitable"
!Interval
|-
!Specific mos interval
! Interval
! Specific mos interval
! Abbreviation
! Abbreviation
!Interval size
! Interval size
|-
|-
| 0-mosstep (unison)
| 0-mosstep (unison)
|Perfect unison
| Perfect unison
|P0ms
| P0ms
|0
| 0
|-
|-
| rowspan="2" |1-mosstep
| rowspan="2" | 1-mosstep
|Minor mosstep (or small mosstep)
| Minor mosstep (or small mosstep)
| m1ms
| m1ms
|s
| s
|-
|-
|Major mosstep (or large mosstep)
| Major mosstep (or large mosstep)
|M1ms
| M1ms
| L
| L
|-
|-
| rowspan="2" |'''2-mosstep'''
| rowspan="2" | '''2-mosstep'''
|Diminished 2-mosstep
| Diminished 2-mosstep
|d2ms
| d2ms
|2s
| 2s
|-
|-
|'''Perfect 2-mosstep'''
| '''Perfect 2-mosstep'''
|P2ms
| P2ms
|L+s
| L + s
|-
|-
| rowspan="2" |3-mosstep
| rowspan="2" | 3-mosstep
|Minor 3-mosstep
| Minor 3-mosstep
|m3ms
| m3ms
|1L+2s
| 1L + 2s
|-
|-
|Major 3-mosstep
| Major 3-mosstep
|M3ms
| M3ms
|2L+s
| 2L + s
|-
|-
| rowspan="2" |4-mosstep
| rowspan="2" | 4-mosstep
|Minor 4-mosstep
| Minor 4-mosstep
|m4ms
| m4ms
|1L+3s
| 1L + 3s
|-
|-
|Major 4-mosstep
| Major 4-mosstep
|M4ms
| M4ms
|2L+2s
| 2L + 2s
|-
|-
| rowspan="2" |'''5-mosstep'''
| rowspan="2" | '''5-mosstep'''
|'''Perfect 5-mosstep'''
| '''Perfect 5-mosstep'''
|P5ms
| P5ms
|2L+3s
| 2L + 3s
|-
|-
|Augmented 5-mosstep
| Augmented 5-mosstep
|A5ms
| A5ms
|3L+2s
| 3L + 2s
|-
|-
| rowspan="2" |6-mosstep
| rowspan="2" | 6-mosstep
|Minor 6-mosstep
| Minor 6-mosstep
|m6ms
| m6ms
|2L+4s
| 2L + 4s
|-
|-
|Major 6-mosstep
| Major 6-mosstep
| M6ms
| M6ms
|3L+3s
| 3L + 3s
|-
|-
|7-mosstep (octave)
| 7-mosstep (octave)
| Perfect octave
| Perfect octave
|P7ms
| P7ms
|3L+4s
| 3L + 4s
|}
|}
== Expanding names for smaller mosses ==
TAMANMS additionally provides optional names for mosses with fewer than six steps. These mosses require that some small integer multiple of the period is equal to an octave, under the reasoning that such step patterns are common and broad in tuning that their names can be validly reused in non-octave contexts. As a result, these names are chosen to be as general as possible, so as to avoid any bias or flavor towards anything other than their step counts or step patterns.
By default, all names assume octave-equivalence or equivalence with a tempered octave. When discussing these scales in a non-octave context, it's recommended to say ''equave''-equivalent ''pattern'' (e.g. "3/1-equivalent tetric" when discussing a 3L&nbsp;1s pattern in an tritave-equivalent context); if context is established, such as if the scale signature is present, this can be dropped and the pattern name by itself can be said.
The exception to this are the names ''monowood'' and ''biwood'', which must refer to an octave-equivalent mos pattern of 1L&nbsp;1s or 2L&nbsp;2s, respectively. Additionally, the name ''monowood'' is advised over ''trivial'' to refer to an octave-equivalent 1L&nbsp;1s scale.
{| class="wikitable center-all"
|-
! colspan="6" | 2-note mosses
|-
! Pattern !! Name !! Prefix !! Abbr.
! Can be non-octave? !! Etymology
|-
| rowspan="2" | [[1L&nbsp;1s]] || trivial || triv- || trv
| Yes || The simplest valid mos pattern.
|-
| monowood
| monowd-
| w
| No (octave-only)
| Blackwood[10] and whitewood[14] generalized to 1 period.
|-
! colspan="6" | 3-note mosses
|-
! Pattern !! Name !! Prefix !! Abbr.
! Can be non-octave? !! Etymology
|-
| [[1L&nbsp;2s]] || antrial || atri- || at
| Yes || Opposite pattern of 2L&nbsp;1s, with broader range. Shortening of ''anti-trial''.
|-
| [[2L&nbsp;1s]] || trial || tri- || t
| Yes || From tri- for 3.
|-
! colspan="6" | 4-note mosses
|-
! Pattern !! Name !! Prefix !! Abbr.
! Can be non-octave? !! Etymology
|-
| [[1L&nbsp;3s]] || antetric || atetra- || att
| Yes || Opposite pattern of 3L&nbsp;1s, with broader range. Shortening of ''anti-tetric''.
|-
| [[2L&nbsp;2s]] || biwood || biwd- || bw
| No (octave-only) || Blackwood[10] and whitewood[14] generalized to 2 periods.
|-
| [[3L&nbsp;1s]] || tetric || tetra- || tt
| Yes || From tetra- for 4.
|-
! colspan="6" | 5-note mosses
|-
! Pattern !! Name !! Prefix !! Abbr.
! Can be non-octave? !! Etymology
|-
| [[1L&nbsp;4s]] || pedal || ped- || pd
| Yes || From Latin ''ped'', for ''foot''; one big toe and four small toes.
|-
| [[2L&nbsp;3s]] || pentic || pent- || pt
| Yes || Common pentatonic; from penta- for 5.
|-
| [[3L&nbsp;2s]] || antipentic || apent- || apt
| Yes || Opposite pattern of 2L&nbsp;3s.
|-
| [[4L&nbsp;1s]] || manual || manu- || mnu
| Yes || From Latin ''manus'', for ''hand''; one thumb and four longer fingers.
|}
== Reasoning for mos pattern names ==
== Reasoning for mos pattern names ==
The goal of TAMNAMS mos names is to choose memorable but aesthetically neutral names.
The goal of TAMNAMS mos names is to choose memorable names for the most common octave-equivalent mosses. Generally, names should befit the mos they're describing ''no matter what temperaments support it'', allowing them to be discussed agnostically of any RTT-related contexts.
 
Names are given to mosses that are the most likely to be used by musicians. As such, TAMNAMS primarily provides names for mosses within the range of 6 to 10 steps (or 2 to 10 steps, when including the names for smaller mosses). This range is chosen to avoid naming large mosses ''for the sake of naming''. Additionally, some of these reasonings also serve as justifications for changing earlier names. As such, this section not only provides reasonings for their names but also a record of how those reasonings were developed.
 
=== General reasonings ===
The following reasonings cover most TAMNAMS names and should be considered the minimum criteria for naming mosses.
 
Notable non-temperament names are incorporated into TAMNAMS if they do not cause confusion, or are given names that reference notable things. Such names include ''mosh'', ''tcherepnin'', ''archaeotonic'', ''oneirotonic'', ''balzano'', ''armotonic'', ''checkertonic'', and ''diatonic.''
 
The name of an interval or a diatonic interval quality can be incorporated into the name of a mos. Such names include ''smitonic'', ''gramitonic'', ''semiquartal'', ''subneutralic'', and ''sinatonic'', from "sharp minor third", "grave minor third", "half-fourth", "between supraminor and neutral", and the interval [[sinaic]], respectively.
 
Temperament-based names ending in the suffix ''-oid'' refer to [[Exotemperament|exotemperaments]] (low-accuracy temperametns) whose tuning ranges, when including extreme tunings, cover the entirety of their corresponding mosses. Therefore, edos with simple step ratios (2:1, 3:1, 3:2, etc) for that mos will correspond to valid tunings for that temperament (if not by [[patent val]], then with a small number of [[warts]]). Such names include ''machinoid'', ''dicoid'', and ''sephiroid'', in reference to [[machine]], [[dichotic]]/[[dicot]], and [[sephiroth]] temperaments, respectively; for information regarding these temperaments' tunings, see their specific reasonings below.
 
Temperament-based names that don't refer to exotemperaments are used ''as a last resort'', and if used should be based on a notable temperament. Most of these names are abstractions of their original temperament names insofar that they refer to a temperament. Such names include ''pine'', ''hyrulic'', ''jaric'', ''ekic'' and ''lemon''; these reference the temperaments of [[porcupine]], [[triforce]], [[pajara]] (along with [[diaschismic]] and [[injera]]), [[echidna]], and [[lemba]], respectively, with ''jaric'' and ''lemon'' having additional reasonings of their own.
 
==== Reasonings for ''n''L&nbsp;''n''s mosses====
Mosses of the form ''n''L&nbsp;''n''s are given names based on a Greek numeral prefix added to the base name ''wood'', in reference to the temperaments [[blackwood]] and [[whitewood]]. These mosses are special in that all mosses with the same number of periods ''n'' can be traced back to an ''n''L&nbsp;''n''s mos, representing a mos consisting of only its generators and periods. In other words, these mosses are a 1L&nbsp;1s pattern repeated ''n'' times in one octave. Coincidentally, all mosses with ''n'' periods form a binary ''tree'' whose ''root'' is ''n''L&nbsp;''n''s (and wood is generally known to come from trees), lending credence to the wood-based name.
 
==== Monolarge mosses ====
[[Step-generated scale|Monolarge]] mosses (mosses of the form 1L&nbsp;''n''s) are given names based on their sister mos (''n''L&nbsp;1s), with the ''anti-'' prefix added. The exception to this is 1L&nbsp;6s, given the name ''onyx'' for the following reasonings:
 
<blockquote>"1Ln-ic's" and "nL1-ic's (like, the -ic suffix applied to MOSS names, collectivised for 1Lns and nL1s) sounds like "one-el-en-ics" or "en-el-one-ics" which abbreviated sort of sounds like "one-ics" => "onyx". Then "onyx" sounds sort of like "one-six". Furthermore the onyx mineral comes in many colours and types, which seems fitting given this is the parent scale for a wide variety of MOSSes; specifically of interest being 7L&nbsp;1s (pine), 8L&nbsp;1s (subneutralic) and 9L&nbsp;1s (sinatonic). Finally, the name "onyx" is also supposed to be vaguely reminiscent of "anti-archaeotonic" as "chi" (the greek letter) is written like an "x" (this is related to why "christmas" is abbreviated sometimes as "X-mas") and other than that, the letters "o" and "n" and their sounds are also present in "archaeotonic", and "x" is vaguely reminiscent of negation and multiplication. There is also something like a "y" sound in "archaeotonic" in the "aeo" part (depending partially on your pronounciation).</blockquote>
 
Monolarge mosses were originally left unnamed due to the tuning ranges for these mosses being so large that they were unhelpful with knowing how they sound. This position was later amended as it's useful for describing structure in situations where one does not want to use the mathematical name, and especially in such contexts, a specific tuning will likely be specified.
 
==== Malic (2L&nbsp;4s), citric (4L&nbsp;2s), lime (4L&nbsp;6s), and lemon (6L&nbsp;4s) ====
The names for 2L&nbsp;4s and 4L&nbsp;2s come from Latin ''malus'' and ''citrus'', meaning 'apple' and 'citrus', respectively. Apples have concave ends, whereas lemons and limes&mdash;both types of citrus fruits&mdash;have convex ends. Both are ubiquitous foods, justifying their use for these fairly small mosses.
 
The name ''citric'' is given to 4L&nbsp;2s, as it is the parent mos of 6L&nbsp;4s and 4L&nbsp;6s, named after the citrus fruits ''lemon'' and ''lime'', respectively, under the reasoning that lemons are larger than limes, as are the step sizes of 6L&nbsp;4s compared to that of 4L&nbsp;6s.
 
Originally, the names for 4L&nbsp;6s and 6L&nbsp;4s were based on the duplication of the 2L&nbsp;3s mos and were called ''dipentic'' and ''antidipentic'', respectively. These were changed to their current names as, at the time, the 5-note mosses required an octave period, thus these names required an equivalence interval of 4/1. Although the name ''pentic'' can currently apply to a 2L&nbsp;3s pattern with any size period, the current names were given for completeness, which warranted renaming the related mosses of 2L&nbsp;4s (renamed from ''antilemon'' to ''malic'') and 4L&nbsp;2s (renamed from ''lemon'' to ''citric'').


All names ending in -oid refer to an exotemperament which, when including extreme tunings, covers the entire range of the corresponding octave-period mos, such that many edos with simple step ratios for that mos will correspond to valid tunings, if not by patent val, then with a small number of warts.
==== Subaric (2L&nbsp;6s), jaric (2L&nbsp;8s), and taric (8L&nbsp;2s) ====
The name ''jaric'' alludes to a few highly notable temperaments that exist in the tuning range of 8L&nbsp;2s, which are alluded to through the spelling and pronunciation of '''jaric''': [[Pajara|pa'''jar'''a]], [[Injera|in'''jer'''a]], and [[Diaschismic|diaschism'''ic''']]. These temperaments, except for diaschismic, have generally inaccurate tunings.


All names for mosses with five or less notes - excluding (mono)wood and biwood (which like all n-wood mosses are specific to octave tuning) - require that some small integer multiple of the period is equal to an octave, under the reasoning that mosses with five or less notes are common and broad in tuning enough that they are much more likely to find interest in non-octave contexts. Because of this, their names were chosen to be extremely general, both to avoid bias/being too flavorful and (correspondingly) so that the terms could validly be reused for any mos for which the period is not equal to a (potentially tempered) octave.
The name ''taric'' was named based on it being the only named-range mos with a basic tuning ({{nowrap|L:s {{=}} 2:1}}) of [[18edo]] and, as it and 2L&nbsp;8s share the same parent of 2L&nbsp;6s, was made to rhyme with jaric.


Any multiperiod mos with more than five notes was given a name that wasn't reliant on the name of a mos with five or less notes as such names were based on those mos names formerly requiring an octave tuning (which is to say some small integer multiple of their period must be equal to a (potentially slightly tempered) octave).
The name ''subaric'' alludes to the fact that 2L&nbsp;6s is the largest proper '''sub'''set mos of both j'''aric''' (2L&nbsp;8s) and t'''aric''' (8L&nbsp;2s).


Former names like "orwelloid" and "sensoid" were abandoned because the names were too temperament-specific in the sense that even considering extreme tunings did not cover the whole range of the mos. The remaining temperament-based names have been abstracted or altered heavily, namely "pine", "hyrulic", "jaric", "ekic" and "lemon".
Originally, the names for 2L&nbsp;8s and 8L&nbsp;2s were based on the duplication of the 3L&nbsp;2s mos and were called called ''antidimanic'' and ''dimanic'', respectively (note that ''manic'' was since changed to ''manual''). These were changed for the same reasons as with 4L&nbsp;6s and 6L&nbsp;4s, and similarly warranted renaming the related mosses of 2L&nbsp;6s (renamed from ''antiechidnoid'' to ''subaric'') and 6L&nbsp;2s (renamed from ''echidnoid'' to ''ekic'').


The inclusion of mos names for "multiperiod" mosses was from a desire to have all ten-note-and-under mosses named for completeness, which is also what prompted some of the reconsiderations mentioned earlier. Similarly, the inclusion of mosses of the form 1L ns using the "anti-" prefix (or an- for less-than-six-note mosses) was also for a practical consideration; although the tuning range is very unhelpful for knowing what such a mos will sound, it is nonetheless useful for describing structure in situations where one does not want to use the mathematical name, especially given that in such situations the tuning will likely be specified somewhere already. Jaric and taric specifically were chosen over bipedal and bimanual because of this, and to a lesser extent, lemon and lime were chosen over antibipentic and bipentic respectively (and for consistency with that their parent MOSS, 4L2s, is named citric).
{| class="wikitable"
|+ style="font-size: 105%;" | Two-period mosses and name changes
|-
! Pattern
! Name
! Pattern
! Name
! Pattern
! Name
! Pattern
! Name
|-
| rowspan="5" | ''2L&nbsp;2s''
| rowspan="5" | biwood<br />''(formerly unnamed)''
| rowspan="2" | 4L&nbsp;2s
| rowspan="2" | citric<br />''(formerly lemon)''
| 4L&nbsp;6s
| lime<br />''(formerly dipentic)''
|
|
|-
| 6L&nbsp;4s
| lemon<br />''(formerly antidipentic)''
|
|
|-
| rowspan="3" | 2L&nbsp;4s
| rowspan="3" | malic<br />''(formerly antilemon)''
| 6L&nbsp;2s
| ekic<br />''(formerly echidnoid)''
|
|
|-
| rowspan="2" | 2L&nbsp;6s
| rowspan="2" | subaric<br />''(formerly antiechidnoid)''
| 8L&nbsp;2s
| taric<br />''(formerly antidimanic)''
|-
| 2L&nbsp;8s
| jaric<br />''(formerly dimanic)''
|}


The distinction between using the prefixes "anti-" vs "an-" for reversing the number of large vs. small steps is also not as trivial as it may sound. In the case of mosses with six or more notes, as the period is always an octave, there is a very large tuning range for the 1L ns mosses (hence the original reason for omitting such mosses), but the "anti-" prefix shows that what is significant is that it has the opposite structure to the corresponding nL 1s mos while pointing out the resulting ambiguity of range. In the case of mosses with five or less notes, as the period is not known and therefore could be very small, this is not as much of a concern as fuller specification is likely required anyway, especially in the case of larger periods, so the name should not be tediously long as the name refers to a very simple mos pattern, and for related reasons, the name shouldn't give as much of a sense of one 'orientation' of the structure being more 'primary' than the other, while with mosses with more than five notes, this suggestion of sense is very much intended, because it will almost always make more sense to talk about the (n+1)L 1s child mos of whatever 1L ns mos you want to speak of.
=== Reasonings for specific names ===
===Name-specific reasonings===
==== [[#Monolarge mosses|Onyx (1L&nbsp;6s)]] ====
====Pedal (1L 4s)====
See [[#Monolarge mosses]].
Pedals are operated with feet, which have one large toe and four small toes. Also comes from words like "bipedal", where in TAMNAMS, "bipedal" would literally mean a pedal scale with a period equal to half of some chosen interval, although such a scale would have either two right feet or two left feet depending on orientation chosen. If you think "car"/"vehicle" when you think "pedal" and don't think (or want to think) much about feet then you can think about "[[beep]]ing" (as [[beep]] is the 7-limit 4&5 exotemperament). Because this name relies so heavily and fundamentally on there being 1 large and 4 small steps per period, it is appropriate to generalise for any size of period you would want. In that regard, same goes for manual, pentic and anpentic.
====Malic (2L 4s) and citric (4L 2s)====
Malic derives from Latin ''malus'' 'apple'. An apple has two concave ends, and large steps in a scale with more small steps are hole-like, hence the two large steps in malic. Citric (4L 2s) is named after the child mosses of citric, namely lemon (6L 4s) and lime (4L 6s). Unlike apples, lemons have two convex pointy ends, and small steps in a scale with more large steps are pointy, hence the two small steps. Malic and citric acids are both ubiquitous in food and biology, thus justifying their use for fairly small mos scales.
====Machinoid (5L 1s)====
[[Machine]] is the 5&amp;6 temperament in the 2.9.7.11 subgroup with a comma list of 64/63 and 99/98.


This temperament is supported by {{Optimal ET sequence| 5, 6, 11, 12, 16, 17, 22, 23, 27, 28 and 33 }} equal divisions, with non-patent val tunings including 5+5=10e, 5+10e+12=21be, 5+5+5+5+6=26qe, which are mentioned here for demonstrating virtual completeness of the tuning range, and the unusually large [[33edo]] tuning being to show [[11edo]]'s strength as a tuning.
==== Machinoid (5L&nbsp;1s)====
====Onyx (1L 6s)====
[[Machine]] is the {{nowrap|5 &amp; 6}} temperament in the 2.9.7.11 subgroup with a comma list of 64/63 and 99/98.
"1Ln-ic's" and "nL1-ic's (like, the -ic suffix applied to MOSS names, collectivised for 1Lns and nL1s) sounds like "one-el-en-ics" or "en-el-one-ics" which abbreviated sort of sounds like "one-ics" => "onyx". Then "onyx" sounds sort of like "one-six". Furthermore the onyx mineral comes in many colours and types, which seems fitting given this is the parent scale for a wide variety of MOSSes; specifically of interest being 7L 1s (pine), 8L 1s (subneutralic) and 9L 1s (sinatonic). Finally, the name "onyx" is also supposed to be vaguely reminiscent of "anti-archaeotonic" as "chi" (the greek letter) is written like an "x" (this is related to why "christmas" is abbreviated sometimes as "X-mas") and other than that, the letters "o" and "n" and their sounds are also present in "archaeotonic", and "x" is vaguely reminiscent of negation and multiplication. There is also something like a "y" sound in "archaeotonic" in the "aeo" part (depending partially on your pronounciation).
====Subaric (2L 6s), jaric (2L 8s), and taric (8L 2s)====
The name "subaric" alludes to the fact that 2L 6s is the largest proper '''sub'''set mos of both j'''aric''' (2L 8s) and t'''aric''' (8L 2s).


The name "jaric" alludes to a few highly notable and generally inaccurate (with the exception of diaschismic) temperaments that exist in the tuning range of this MOSS. Specifically, notice how the letters and sound of "jaric" has (or is intended to have) a lot of overlap with [[pajara|pa'''jar'''a]], [[diaschismic|diaschism'''ic''']] and [[injera|in'''jer'''a]] (listed in order of increasingly sharp fourths; note that diatonic fourths and 4-jarasteps are equated in jaric, a notable property).
This temperament is supported by {{Optimal ET sequence| 5, 6, 11, 12, 16, 17, 22, 23, 27, 28 and 33 }} equal divisions, many of which correspond to both simple tunings ({{nowrap|L:s {{=}} 2:1}}, 3:1, 3:2, etc) and degenerate tunings ({{nowrap|L:s {{=}} 1:1}} or 1:0) for 5L&nbsp;1s. Non-patent val tunings include {{nowrap|5 + 5 {{=}} 10e|5 + 10e + 12 {{=}} 21be|and 5 + 5 + 5 + 5 + 6 {{=}} 26qe}}; these are mentioned here for demonstrating virtual completeness of the tuning range, as is 33edo to show 11edo's strength as a tuning.


The name "taric" was named based on it being the only octave-tuned TAMNAMS pattern with a [[#Simple step ratios|basic]] tuning of [[18edo]] (because [[7L 4s]] has more than 10 notes so is out of the scope of TAMNAMS, although not necessarily out of the scope of extensions) and it was also named based on rhyming with jaric (as they share the parent mos [[2L 6s]]).
==== Sephiroid (3L&nbsp;7s) ====
====Sephiroid (3L 7s)====
[[Sephiroth]] is the {{nowrap|3 &amp; 10}} temperament in the 2.5.11.13.17.21 subgroup with commas including 65/64, 85/84, 105/104, 169/168, 170/169, 221/220, 273/272, 275/273.
[[Sephiroth]] is the 3&amp;10 temperament in the 2.5.11.13.17.21 subgroup with commas including 65/64, 85/84, 105/104, 169/168, 170/169, 221/220, 273/272, 275/273.


This temperament is supported by {{Optimal ET sequence| 3, 10, 13, 16, 23 and 26 }} equal divisions, with non-patent val tunings including 6eg, 7e*, 19eg, 20e, 29g, 32egq, 33ce, 36c.
This temperament is supported by {{Optimal ET sequence| 3, 10, 13, 16, 23, and 26 }} equal divisions, with non-patent val tunings including 6eg, 7e, 19eg, 20e, 29g, 32egq, 33ce, 36c. Like with that of 5L&nbsp;1s, these represent both simple and degenerate tunings for 3L&nbsp;7s. Extreme tunings, such as 7e, may lie outside the mos's step ratio spectrum, although such tunings are generally not considered good tunings.


<nowiki>*</nowiki> Extreme tunings even occasionally go outside of this range like with 7e, but this would never be considered a good tuning.
==== Dicoid (7L&nbsp;3s) ====
[[Dicot family#Dichotic|Dichotic]] is the {{nowrap|7 &amp; 10}} temperament in the 11-limit with commas including 25/24, 45/44, 55/54, 56/55, 64/63. This is an extension of the 5-limit exotemperament [[dicot]] which tempers 25/24, equating 5/4 and 6/5 into a neutral third sized interval, which is the generator.


(Note that ''q'' in the above is a placeholder symbol meaning that the generator 21 is warted.)
This temperament is supported by {{Optimal ET sequence| 7, 10, and 17 }} equal divisions, with non-patent val tunings including (but not limited to) {{nowrap|7 + 7 {{=}} 14cd|10 + 10 {{=}} 20e|17 + 7 {{=}} 24cd|and 17 + 10 {{=}} 27ce}}.


Note therefore how practically a full range of tunings is covered both in breadth and depth.
==== Armotonic (7L&nbsp;2s) ====
====Dicoid (7L 3s)====
Originally, the name ''superdiatonic'' was used for 7L&nbsp;2s, as it has seen some precedent of use on the wiki to refer to an octave-equivalent 7L&nbsp;2s pattern, although it has had earlier use to refer to the expansion of a smaller mos to a larger one. Due to these concerns, the name ''armotonic'' is normally advised over ''superdiatonic'' as the former is unambiguous as to what it refers to, and the name ''superdiatonic'' is only allowed in situations where it's truly unambiguous if the writer prefers it.
[[Dicot family#Dichotic|Dichotic]] is the 7&amp;10 temerament in the 11-limit with commas including 25/24, 45/44, 55/54, 56/55, 64/63 and is an extension of the 5-limit exotemperament [[dicot]] which tempers 25/24, equating 5/4 and 6/5 into a neutral third sized interval, which is the generator. To help justify using these temperament for inspiration for the name, note that:


This temperament is supported by {{Optimal ET sequence| 7, 10 and 17 }} equal divisions, with non-patent val tunings including 14cd(=7+7), 20e(=10+10), 24cd(=17+7), 27ce(=17+10).
==== On the term ''diatonic'' ====
Although the term ''diatonic'' has accrued a variety of exact meanings over time, both within and outside the contexts of xenharmonic music theory, in the context of TAMNAMS and moment-of-symmetry scales, the term ''diatonic'' exclusively refers to 5L&nbsp;2s.


Note there are many more warted tunings than this with even more extreme tunings, which makes it reasonable to loosely associate the exotemperament with the range of vaguely saner tunings.
== Notes ==
====Superdiatonic/armotonic (7L 2s)====
<references group="note" />
Though the term has seen some use in other contexts, the name "superdiatonic" has seen some precedent of use on the Xenwiki to refer to the mos pattern 7L 2s. This mos is part of a series of mos patterns (5+2k)L 2s, which starts with diatonic (5L 2s, k=0) and superdiatonic (7L 2s, k=1). Like 5L 2s, 7L 2s is also a fifth-generated scale and has a structure similar to diatonic in some ways, but with more large steps. In contexts where the term "superdiatonic" conveys a different meaning (other than referring to 7L 2s), the name "armotonic", in reference to Armodue theory, can be used as an alternative name.
====On the term ''diatonic''====
In TAMNAMS, ''diatonic'' exclusively refers to 5L 2s. Other diatonic-based scales (specifically with 3 step sizes or more), such as [[Zarlino]], [[blackdye]] and [[diasem]], are called ''[[Detempering|detempered]]'' (if the philosophy is [[RTT]]-based) or ''deregularized/detuned'' (RTT-agnostic) ''diatonic scales''. The adjectives ''diatonic-like'' or ''quasi-diatonic'' may also be used to refer to diatonic-based scales, depending on what's contextually the most appropriate.


(The choice of how to define ''diatonic'' isn't bound by history, since many other terms have different meanings depending on the historical musical system referred to, for example the ''enharmonic scale'' may refer to a chain-of-fifths-based scale with 12edo enharmonics not equated, or to an ancient Greek genus.)
[[Category:TAMNAMS]]

Latest revision as of 15:39, 17 April 2025

< TAMNAMS

This page is an appendix to TAMNAMS.

Reasoning for step ratio names

Derivation of the step ratio names

The idea is to start with the simplest ratios (L/s = 1/0 and L/s = 1/1) and derive more complex ratios through repeated application of the mediant (aka Farey addition) to adjacent fractions.

  • Applying the mediant to the starting intervals 1/0 and 1/1 gives (1 + 1)/(1 + 0) = 2/1, and as this is the simplest possible ratio where the large and small step are distinguished and nonzero, it is called the basic tuning. (Note that if applying the mediant to 1/0 seems confusing, think of it as equivalent to applying the mediant to 0/1 and 1/1 and the ratios as flipped, thus representing s/L rather than L/s when written this way.)
  • As L/s = 1/1 represents L and s being equal in size, it is called equalized.
  • As L/s = 1/0 represents s = 0, it is called collapsed, as the small scale steps collapse to zero cents and disappear.
  • The mediant of 1/1 and 2/1 is 3/2, thus making the scale sound mellower/softer, and as this is the simplest (in the sense of lowest integer limit) ratio to represent such a property, it is simply called the soft tuning.
  • Analogously, the mediant of 2/1 and 1/0, 3/1, is called the hard tuning. Thus you can say that a step ratio tuning is hard of or soft of another step ratio tuning.
  • To get something between soft and basic we take the mediant again and get 5/3 for semisoft, and analogously 5/2 for semihard. To get something more extreme we take the mediant of 1/0 with 3/1 for a harder-than-hard tuning, giving us 4/1 for superhard and analogously 4/3 for supersoft.

There are also tertiary names beyond the above:

  • Anything softer than supersoft is ultrasoft, and anything harder than superhard is ultrahard. Something between soft and supersoft is parasoft, as para- means both beyond and next to. Something between hard and superhard is parahard.
  • Something between soft and basic is hyposoft as it is less soft than soft. Something between hard and basic is hypohard for the same reason. Between semisoft and basic is minisoft and between semihard and basic is minihard.
  • Finally, between soft and semisoft is quasisoft as such scales may potentially be mistaken for soft or semisoft while not being either—hence the use of the prefix quasi-, and between hard and semihard is quasihard for the same reason.

The reasoning for the para-/super-/ultra- progression (note that super- is the odd one out as it refers to an exact ratio) is it mirrors naming for shades of musical intervals and because parapythagorean is between pythagorean and superpythagorean.

This results in the central spectrum—an elegant system which names all exact L/s ratios in the 5-integer-limit excepting only 5/1 and 5/4 which are disincluded intentionally for a variety of reasons: to keep the maximum corresponding notes per period in an equal pitch division low, because it keeps the 'tree' of mediants complete to a certain number of layers, and because their disinclusion gives a roughly-equally-spaced set of ratios, with the regions between 4/3 and 1/1 and between 4/1 and 1/0 being the only exceptions—corresponding to extreme tunings. Note that filling in those extreme regions is the purpose of the extended spectrum

Extending the spectrum's edges

Extending the spectrum builds on the central spectrum and relies on a few key observations.

Firstly, as periods and mosses come in wildly different shapes and sizes, and as we want to represent a somewhat representative variety of simple tunings for the step ratio for a given mos pattern and period, the notion of simple used will correspond to the number of equally-spaced tones per period required. This is expressed as xL + ys, where x and y are the number of large and small steps in the scale, and where L and s are from the step ratio L/s (where L and s are assumed to be coprime). Then, in order to not introduce bias to mos patterns with more L's or more s's, we should assume that both are equally likely and thus weight both equally, which means that the resulting minimum number of tones per period for a ratio L/s is L + s.

The next observation is that the large values of L/s can be a lot more consequential than the ones close to 1/1 due to the fact that small steps are guaranteed to be smaller than large steps and that we don't know how many small steps there are compared to large steps, and therefore the hard end of the spectrum is more vast, and analogously, L/s values close to 1/1 will tend to be inconsequential and for very close values likely impractical to distinguish—in the extremes only serving small tuning adjustments rather than melodic properties. This leads to another observation: mos patterns with periods tuned to step ratios, while related to temperaments, are not temperaments—instead forming a sort of amalgamative superset of temperaments if you want to force a temperament interpretation, and thus their main function is in melodic structure, with temperaments informing potential harmonies and microtunings. Thus, the spectrum should be kept minimal and simple so that it is both generally hearable and not too specific.

The most obvious adjustment to the edges is to draw a distinction between ultrasoft and pseudoequalized by adding a step ratio corresponding to semiequalized, and between ultrahard and pseudocollapsed by adding a step ratio corresponding to semicollapsed. Thus:

Ultrasoft is between supersoft and semiequalized and pseudoequalized is between semiequalized and equalized.

Ultrahard is between superhard and semicollapsed, and pseudocollapsed is between semicollapsed and collapsed.

Then all that's left is to decide what the step ratios for semicollapsed and semiequalized should be. In order to keep the spacing (of the s/L ratios when graphed, or to a lesser extent the L/s ratios if you see the roughly gradual increase in spacing in that form) roughly consistent with all the other ratios, semiequalized should be L/s = 6/5 rather than L/s = 5/4. Then note the complexity of L/s = 6/5 is 6 + 5 = 11, so to find the corresponding complexity for semicollapsed we use L/s = 10/1 as 10 + 1 = 11 too. Then finally, to preserve some of the symmetry, we include L/s = 6/1 as extrahard. Although L/s = 10/1 for semicollapsed may seem a little extreme of a boundary, L/s = 12/1 would actually be what is the most equally spaced continuing on from 6/1 for the same reason that L/s = 6/5 is the most equally spaced. Note that while the range from superhard to semicollapsed is ultrahard, the region may be split into two sub-ranges:

superhard (L/s = 4/1) to extrahard (L/s = 6/1) is hyperhard (4 < L/s < 6).

extrahard (L/s = 6/1) to semicollapsed (L/s = 10/1) is clustered (6 < L/s < 10).

With the inclusion of these 3 new L/s ratios nearer the edges of the spectrum and names for the range divisions they create, we get the extended spectrum, summarised and detailed above, just for the regions affected to avoid repetition.

Extended spectrum

Extended spectrum of step ratio ranges and specific step ratios
Central ranges Extended ranges Specific
step ratios 		Notes
1:1
(equalized) 		Trivial/pathological
1:1 to 2:1
(soft-of-basic) 		1:1 to 4:3
(ultrasoft) 		1:1 to 6:5
(pseudoequalized)
6:5
(semiequalized)
6:5 to 4:3
(ultrasoft)
4:3
(supersoft) 		Nonextreme range, as detailed
by central spectrum
4:3 to 3:2
(parasoft) 		4:3 to 3:2
(parasoft)
3:2
(soft)
3:2 to 2:1
(hyposoft) 		3:2 to 5:3
(quasisoft) 		3:2 to 5:3
(quasisoft)
5:3
(semisoft)
5:3 to 2:1
(minisoft) 		5:3 to 2:1
(minisoft)
2:1
(basic)
2:1 to 1:0
(hard-of-basic) 		2:1 to 3:1
(hypohard) 		2:1 to 5:2
(minihard) 		2:1 to 5:2
(minihard)
5:2
(semihard)
5:2 to 3:1
(quasihard) 		5:2 to 3:1
(quasihard)
3:1
(hard)
3:1 to 4:1
(parahard) 		3:1 to 4:1
(parahard)
4:1
(superhard)
4:1 to 1:0
(ultrahard) 		4:1 to 10:1
(ultrahard) 		4:1 to 6:1
(hyperhard)
6:1
(extrahard)
6:1 to 10:1
(clustered)
10:1
(semicollapsed)
10:1 to 1:0
(pseudocollapsed)
1:0
(collapsed) 		Trivial/pathological

Terminology and final notes

A ratio of L/s = k/1 can be called k-hard and a ratio of L/s = k/(k − 1) can analogously be called k-soft, so the simplest ultrasoft tuning is 5-soft or pentasoft, the simplest hyperhard tuning is 5-hard or pentahard, the simplest clustered tuning is 7-hard or heptahard, 8-hard is octahard, 9-hard is nonahard, and finally, the characteristic simple ultrahard tuning is 6-hard or extrahard, as previously discussed, which can be seen to be similar to hexahard—hopefully helping with memorisation.

A perhaps useful (or otherwise mildly amusing) mnemonic is 2-soft is too soft to be hard and 2-hard is too hard to be soft, representing that 2-soft = 2-hard = 2/1 = basic.

Note that often the central spectrum will be sufficient for exploring a mos pattern-period combination, and the extended spectrum is intended more for (literally) edge cases where it may be useful. Often if a temperament interpretation doesn't seem to show up for a mos pattern-period combination, it just means the temperament needs a more complex mos pattern to narrow down the generator range. An example of this phenomena is the highly complex mos pattern of 12L 17s represents near-Pythagorean tunings well due to having a generator of a fourth or a fifth bounded between those of 12edo and those of 29edo, which are roughly equally off but in opposite directions, and many important near-Pythagorean systems show up in just the ratios of the central spectrum alone.

Reasoning for mos interval names

Reasoning for 0-indexed intervals

Note that a unison is a 0-mosstep, rather than a mos1st; likewise, the term 1-mosstep is used rather than a mos2nd. One might be tempted to generalize diatonic 1-indexed ordinal names: In 31edo's ultrasoft mosh scale, the perfect mosthird (aka Pmosh3rd) is a neutral third and the major mosfifth (aka Lmosh5th) is a perfect fifth. The way intervals are named above (and in 12edo theory) has a problem. An interval that's n steps wide is named (n+1)th. This means that adding two intervals is more complicated than it should be. Stacking two fifths makes a ninth, when naively it would make a tenth. We're used to this for the diatonic scale, but when dealing with unfamiliar scale structures, it can be very confusing.

To overcome this, TAMNAMS uses a 0-indexed name system for non-diatonic mos intervals, which makes the arithmetic needed to understand mos intervals much smoother. Going up a 0-mosstep means to go up zero steps, and stacking two 4-mossteps produces an 8-mosstep, rather than stacking two mos5ths to produce a mos9th. The use of ordinal indexing is generally discouraged when referring to non-diatonic mos intervals.

Deriving the names for mossteps

To find what mos interval sizes are found in a mos, start with the patterns of large and small steps that represents the mos in its brightest mode and its darkest mode (which is the reverse pattern for the mos's brightest mode). The idea of mos recursion may be of help with finding the generators of a mos. Likewise, the idea of modal brightness and UDP may be of help for a mos's modes.

Given the mos xL ys, the following algorithm is used to find the brightest mode for that mos.

  1. If either x or y is equal to 1 (base cases):
    1. If both x and y are equal to 1, then the final scale is "Ls".
    2. If only x is equal to 1, then the final scale is L followed by y s's.
    3. If only y is equal to 1, then the final scale is x L's followed by s.
  2. If neither x nor y is equal to 1 (recursive cases):
    1. Let k be the greatest common factor of x and y.
    2. If x and y share a common factor k, where k is greater than 1, then recursively call this algorithm to find the scale for (x/k)L (y/k)s; the final scale will be (x/k)L (y/k)s duplicated k times.
    3. If x and y don't share a common factor that is greater than 1 (if x and y are coprime), then:
      1. Let m1 = min(x, y) and m2 = max(x, y).
      2. Let z = m2 mod m1 and w = m1 − z.
      3. Let prescale be the mos string for zL ws. Recursively call this algorithm to find the scale for zL ws; the final scale will be based on this.
      4. If x < y, reverse the order of characters in the prescale. This is only needed if there are more L's than s's in the final scale.
      5. To produce the final scale, the L's and s's of the prescale must be replaced with substrings consisting of L's and s's. Let u = ⌈m2/m1⌉ and v = ⌊m2/m1⌋.[note 1]
        1. If x > y, every instance of an L in prescale is replaced with one L and u s's, and every s replaced with one L and v s's. This produces the final scale in its brightest mode.
        2. If x < y, every instance of an L in prescale is replaced with u L's and one s, and every s replaced with v L's and one s. This produces the final scale in its brightest mode.

Using 3L 4s as an example, this is LsLsLss (brightest) and ssLsLsL (darkest). To find the large sizes of each k-mosstep, consider the first k mossteps that make up the mos pattern for the brightest mode. Repeat this process with the mos pattern for the darkest mode to find each k-mosstep's small size. To make these sizes more clear, we can denote the mos intervals as a sum of large and small steps iL + js, where i and j are the number of L's and s's in the interval's step pattern; this is to say that the order of L's and s's doesn't matter, rather the amount of each step size. The large and small sizes should differ by replacing one L in the large size with an s.

Interval Large size (LsLsLss) Small size (ssLsLsL)
Step pattern Sum Step pattern Sum
0-mosstep (unison) none 0 none 0
1-mosstep L L s s
2-mosstep Ls L + s ss 2s
3-mosstep LsL 2L + s ssL 1L + 2s
4-mosstep LsLs 2L + 2s ssLs 1L + 3s
5-mosstep LsLsL 3L + 2s ssLsL 2L + 3s
6-mosstep LsLsLs 3L + 3s ssLsLs 2L + 4s
7-mosstep (octave) LsLsLss 3L + 4s ssLsLsL 3L + 4s

Given the mos xL ys, the following algorithm is used to find the bright generator and its complement.

  1. If either x or y is equal to 1 (base cases):
    1. If both x and y are equal to 1, then the generator is "L" and its complement is "s".
    2. If only x is equal to 1, then the generator is "L" followed by y − 1 s's, and the complement is "s".
    3. If only y is equal to 1, then the generator is "L" and the complement is x − 1 L's followed by "s".
  2. If neither x nor y is equal to 1 (recursive cases):
    1. Let k be the greatest common factor of x and y.
    2. If x and y share a common factor k, where k is greater than 1, then recursively call this algorithm to find the generator and complement for (x/k)L (y/k)s; the intervals returned this way will apply to the period rather than the octave.
    3. If x and y don't share a common factor that is greater than 1 (if x and y are coprime), then:
      1. Let m1 = min(x, y) and m2 = max(x, y).
      2. Let z = m2 mod m1 and w = m1 − z.
      3. Let gen be the scale's generator and comp be the generator's octave complement for the mos zL ws. Recursively call this algorithm to find these intervals for zL ws; the final scale's generator and complement will be based on this.
      4. If x < y, reverse the order of characters in gen and comp, then swap gen and comp. This is only needed if there are more L's than s's in the scale.
      5. To produce the scale's generator and complement, the L's and s's of both intervals must be replaced with substrings consisting of L's and s's. Let u = ⌈m2/m1⌉ and v = ⌊m2/m1⌋.[note 1]
        1. If x > y, every instance of an L in both intervals is replaced with one L and u s's, and every s replaced with one L and v s's. This produces the final scale's generator and complement.
        2. If x < y, every instance of an L in both intervals is replaced with u L's and one s, and every s replaced with v L's and one s. This produces the final scale's generator and complement.

The length of gen is the number of mossteps for the bright generator, and the length of comp is the number of mossteps in the dark generator. For our example of 3L 4s, the algorithm returns the step pattern Ls as the bright generator and LsLss as its complement, which are 2 and 5 mossteps wide, respectively. Since the large size of a bright generator is perfect and its small size diminished, and the large size of a dark generator is augmented and its small size perfect, the scale's generators can be identified as shown in the table.

Interval Specific mos interval Abbreviation Interval size
0-mosstep (unison) Perfect unison P0ms 0
1-mosstep Minor mosstep (or small mosstep) m1ms s
Major mosstep (or large mosstep) M1ms L
2-mosstep Diminished 2-mosstep d2ms 2s
Perfect 2-mosstep P2ms L + s
3-mosstep Minor 3-mosstep m3ms 1L + 2s
Major 3-mosstep M3ms 2L + s
4-mosstep Minor 4-mosstep m4ms 1L + 3s
Major 4-mosstep M4ms 2L + 2s
5-mosstep Perfect 5-mosstep P5ms 2L + 3s
Augmented 5-mosstep A5ms 3L + 2s
6-mosstep Minor 6-mosstep m6ms 2L + 4s
Major 6-mosstep M6ms 3L + 3s
7-mosstep (octave) Perfect octave P7ms 3L + 4s

Expanding names for smaller mosses

TAMANMS additionally provides optional names for mosses with fewer than six steps. These mosses require that some small integer multiple of the period is equal to an octave, under the reasoning that such step patterns are common and broad in tuning that their names can be validly reused in non-octave contexts. As a result, these names are chosen to be as general as possible, so as to avoid any bias or flavor towards anything other than their step counts or step patterns.

By default, all names assume octave-equivalence or equivalence with a tempered octave. When discussing these scales in a non-octave context, it's recommended to say equave-equivalent pattern (e.g. "3/1-equivalent tetric" when discussing a 3L 1s pattern in an tritave-equivalent context); if context is established, such as if the scale signature is present, this can be dropped and the pattern name by itself can be said.

The exception to this are the names monowood and biwood, which must refer to an octave-equivalent mos pattern of 1L 1s or 2L 2s, respectively. Additionally, the name monowood is advised over trivial to refer to an octave-equivalent 1L 1s scale.

2-note mosses
Pattern Name Prefix Abbr. Can be non-octave? Etymology
1L 1s trivial triv- trv Yes The simplest valid mos pattern.
monowood monowd- w No (octave-only) Blackwood[10] and whitewood[14] generalized to 1 period.
3-note mosses
Pattern Name Prefix Abbr. Can be non-octave? Etymology
1L 2s antrial atri- at Yes Opposite pattern of 2L 1s, with broader range. Shortening of anti-trial.
2L 1s trial tri- t Yes From tri- for 3.
4-note mosses
Pattern Name Prefix Abbr. Can be non-octave? Etymology
1L 3s antetric atetra- att Yes Opposite pattern of 3L 1s, with broader range. Shortening of anti-tetric.
2L 2s biwood biwd- bw No (octave-only) Blackwood[10] and whitewood[14] generalized to 2 periods.
3L 1s tetric tetra- tt Yes From tetra- for 4.
5-note mosses
Pattern Name Prefix Abbr. Can be non-octave? Etymology
1L 4s pedal ped- pd Yes From Latin ped, for foot; one big toe and four small toes.
2L 3s pentic pent- pt Yes Common pentatonic; from penta- for 5.
3L 2s antipentic apent- apt Yes Opposite pattern of 2L 3s.
4L 1s manual manu- mnu Yes From Latin manus, for hand; one thumb and four longer fingers.

Reasoning for mos pattern names

The goal of TAMNAMS mos names is to choose memorable names for the most common octave-equivalent mosses. Generally, names should befit the mos they're describing no matter what temperaments support it, allowing them to be discussed agnostically of any RTT-related contexts.

Names are given to mosses that are the most likely to be used by musicians. As such, TAMNAMS primarily provides names for mosses within the range of 6 to 10 steps (or 2 to 10 steps, when including the names for smaller mosses). This range is chosen to avoid naming large mosses for the sake of naming. Additionally, some of these reasonings also serve as justifications for changing earlier names. As such, this section not only provides reasonings for their names but also a record of how those reasonings were developed.

General reasonings

The following reasonings cover most TAMNAMS names and should be considered the minimum criteria for naming mosses.

Notable non-temperament names are incorporated into TAMNAMS if they do not cause confusion, or are given names that reference notable things. Such names include mosh, tcherepnin, archaeotonic, oneirotonic, balzano, armotonic, checkertonic, and diatonic.

The name of an interval or a diatonic interval quality can be incorporated into the name of a mos. Such names include smitonic, gramitonic, semiquartal, subneutralic, and sinatonic, from "sharp minor third", "grave minor third", "half-fourth", "between supraminor and neutral", and the interval sinaic, respectively.

Temperament-based names ending in the suffix -oid refer to exotemperaments (low-accuracy temperametns) whose tuning ranges, when including extreme tunings, cover the entirety of their corresponding mosses. Therefore, edos with simple step ratios (2:1, 3:1, 3:2, etc) for that mos will correspond to valid tunings for that temperament (if not by patent val, then with a small number of warts). Such names include machinoid, dicoid, and sephiroid, in reference to machine, dichotic/dicot, and sephiroth temperaments, respectively; for information regarding these temperaments' tunings, see their specific reasonings below.

Temperament-based names that don't refer to exotemperaments are used as a last resort, and if used should be based on a notable temperament. Most of these names are abstractions of their original temperament names insofar that they refer to a temperament. Such names include pine, hyrulic, jaric, ekic and lemon; these reference the temperaments of porcupine, triforce, pajara (along with diaschismic and injera), echidna, and lemba, respectively, with jaric and lemon having additional reasonings of their own.

Reasonings for nL ns mosses

Mosses of the form nL ns are given names based on a Greek numeral prefix added to the base name wood, in reference to the temperaments blackwood and whitewood. These mosses are special in that all mosses with the same number of periods n can be traced back to an nL ns mos, representing a mos consisting of only its generators and periods. In other words, these mosses are a 1L 1s pattern repeated n times in one octave. Coincidentally, all mosses with n periods form a binary tree whose root is nL ns (and wood is generally known to come from trees), lending credence to the wood-based name.

Monolarge mosses

Monolarge mosses (mosses of the form 1L ns) are given names based on their sister mos (nL 1s), with the anti- prefix added. The exception to this is 1L 6s, given the name onyx for the following reasonings:

"1Ln-ic's" and "nL1-ic's (like, the -ic suffix applied to MOSS names, collectivised for 1Lns and nL1s) sounds like "one-el-en-ics" or "en-el-one-ics" which abbreviated sort of sounds like "one-ics" => "onyx". Then "onyx" sounds sort of like "one-six". Furthermore the onyx mineral comes in many colours and types, which seems fitting given this is the parent scale for a wide variety of MOSSes; specifically of interest being 7L 1s (pine), 8L 1s (subneutralic) and 9L 1s (sinatonic). Finally, the name "onyx" is also supposed to be vaguely reminiscent of "anti-archaeotonic" as "chi" (the greek letter) is written like an "x" (this is related to why "christmas" is abbreviated sometimes as "X-mas") and other than that, the letters "o" and "n" and their sounds are also present in "archaeotonic", and "x" is vaguely reminiscent of negation and multiplication. There is also something like a "y" sound in "archaeotonic" in the "aeo" part (depending partially on your pronounciation).

Monolarge mosses were originally left unnamed due to the tuning ranges for these mosses being so large that they were unhelpful with knowing how they sound. This position was later amended as it's useful for describing structure in situations where one does not want to use the mathematical name, and especially in such contexts, a specific tuning will likely be specified.

Malic (2L 4s), citric (4L 2s), lime (4L 6s), and lemon (6L 4s)

The names for 2L 4s and 4L 2s come from Latin malus and citrus, meaning 'apple' and 'citrus', respectively. Apples have concave ends, whereas lemons and limes—both types of citrus fruits—have convex ends. Both are ubiquitous foods, justifying their use for these fairly small mosses.

The name citric is given to 4L 2s, as it is the parent mos of 6L 4s and 4L 6s, named after the citrus fruits lemon and lime, respectively, under the reasoning that lemons are larger than limes, as are the step sizes of 6L 4s compared to that of 4L 6s.

Originally, the names for 4L 6s and 6L 4s were based on the duplication of the 2L 3s mos and were called dipentic and antidipentic, respectively. These were changed to their current names as, at the time, the 5-note mosses required an octave period, thus these names required an equivalence interval of 4/1. Although the name pentic can currently apply to a 2L 3s pattern with any size period, the current names were given for completeness, which warranted renaming the related mosses of 2L 4s (renamed from antilemon to malic) and 4L 2s (renamed from lemon to citric).

Subaric (2L 6s), jaric (2L 8s), and taric (8L 2s)

The name jaric alludes to a few highly notable temperaments that exist in the tuning range of 8L 2s, which are alluded to through the spelling and pronunciation of jaric: pajara, injera, and diaschismic. These temperaments, except for diaschismic, have generally inaccurate tunings.

The name taric was named based on it being the only named-range mos with a basic tuning (L:s = 2:1) of 18edo and, as it and 2L 8s share the same parent of 2L 6s, was made to rhyme with jaric.

The name subaric alludes to the fact that 2L 6s is the largest proper subset mos of both jaric (2L 8s) and taric (8L 2s).

Originally, the names for 2L 8s and 8L 2s were based on the duplication of the 3L 2s mos and were called called antidimanic and dimanic, respectively (note that manic was since changed to manual). These were changed for the same reasons as with 4L 6s and 6L 4s, and similarly warranted renaming the related mosses of 2L 6s (renamed from antiechidnoid to subaric) and 6L 2s (renamed from echidnoid to ekic).

Two-period mosses and name changes
Pattern Name Pattern Name Pattern Name Pattern Name
2L 2s biwood
(formerly unnamed) 		4L 2s citric
(formerly lemon) 		4L 6s lime
(formerly dipentic)
6L 4s lemon
(formerly antidipentic)
2L 4s malic
(formerly antilemon) 		6L 2s ekic
(formerly echidnoid)
2L 6s subaric
(formerly antiechidnoid) 		8L 2s taric
(formerly antidimanic)
2L 8s jaric
(formerly dimanic)

Reasonings for specific names

Onyx (1L 6s)

See #Monolarge mosses.

Machinoid (5L 1s)

Machine is the 5 & 6 temperament in the 2.9.7.11 subgroup with a comma list of 64/63 and 99/98.

This temperament is supported by 5, 6, 11, 12, 16, 17, 22, 23, 27, 28 and 33 equal divisions, many of which correspond to both simple tunings (L:s = 2:1, 3:1, 3:2, etc) and degenerate tunings (L:s = 1:1 or 1:0) for 5L 1s. Non-patent val tunings include 5 + 5 = 10e, 5 + 10e + 12 = 21be, and 5 + 5 + 5 + 5 + 6 = 26qe; these are mentioned here for demonstrating virtual completeness of the tuning range, as is 33edo to show 11edo's strength as a tuning.

Sephiroid (3L 7s)

Sephiroth is the 3 & 10 temperament in the 2.5.11.13.17.21 subgroup with commas including 65/64, 85/84, 105/104, 169/168, 170/169, 221/220, 273/272, 275/273.

This temperament is supported by 3, 10, 13, 16, 23, and 26 equal divisions, with non-patent val tunings including 6eg, 7e, 19eg, 20e, 29g, 32egq, 33ce, 36c. Like with that of 5L 1s, these represent both simple and degenerate tunings for 3L 7s. Extreme tunings, such as 7e, may lie outside the mos's step ratio spectrum, although such tunings are generally not considered good tunings.

Dicoid (7L 3s)

Dichotic is the 7 & 10 temperament in the 11-limit with commas including 25/24, 45/44, 55/54, 56/55, 64/63. This is an extension of the 5-limit exotemperament dicot which tempers 25/24, equating 5/4 and 6/5 into a neutral third sized interval, which is the generator.

This temperament is supported by 7, 10, and 17 equal divisions, with non-patent val tunings including (but not limited to) 7 + 7 = 14cd, 10 + 10 = 20e, 17 + 7 = 24cd, and 17 + 10 = 27ce.

Armotonic (7L 2s)

Originally, the name superdiatonic was used for 7L 2s, as it has seen some precedent of use on the wiki to refer to an octave-equivalent 7L 2s pattern, although it has had earlier use to refer to the expansion of a smaller mos to a larger one. Due to these concerns, the name armotonic is normally advised over superdiatonic as the former is unambiguous as to what it refers to, and the name superdiatonic is only allowed in situations where it's truly unambiguous if the writer prefers it.

On the term diatonic

Although the term diatonic has accrued a variety of exact meanings over time, both within and outside the contexts of xenharmonic music theory, in the context of TAMNAMS and moment-of-symmetry scales, the term diatonic exclusively refers to 5L 2s.

Notes

  1. ↑ 1.0 1.1 ⌈ ⌉ denotes the ceiling function and ⌊ ⌋ denotes the floor function.
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